Old Testament "Offensive" Prayers:
10 Make them bear their guilt, O God; let them fall by their own counsels; because of the abundance of their transgressions cast them out, for they have rebelled against you. [Psa 5:10 ESV]
This is a prayer that David prayed in the Old Testament.  This is most certainly NOT a defensive prayer.  David is asking God to bring about the destruction of other human beings.  Does that mean I can go on the offensive also and pray this about the violent who defy the law, about those who would undermine anyone and everyone to feed their own greed for money and power?  Can I pray that God will bring about the sudden and forever downfall of corrupt politicians at all levels of government?  Can I pray it even though I am not a King?  I struggled with these questions in view of the New Testament injunction that we love our neighbor - even more, it says to love our enemies.  As I dug into this, I noted that the MacArthur Study Bible (MSB) says that David prayed this in parallel with God's revealed standard, which those he is talking about were violating.  That would say that as long as we are praying in accordance with God's laws, and asking Him to take action rather than taking vengeance ourselves, we can all pray for the destruction of the unjust, here, now, today, in this world.

A similar prayer, much further into Psalms:
19 Oh that you would slay the wicked, O God! O men of blood, depart from me! 20 They speak against you with malicious intent; your enemies take your name in vain. 21 Do I not hate those who hate you, O LORD? And do I not loathe those who rise up against you? 22 I hate them with complete hatred; I count them my enemies. [Psa 139:19-22 ESV]  This is David also.  David is a man after God's own heart.

This next seems about the ultimate example of prayer as an offensive weapon:
6 Appoint a wicked man against him; let an accuser stand at his right hand. 7 When he is tried, let him come forth guilty; let his prayer be counted as sin! 8 May his days be few; may another take his office! 9 May his children be fatherless and his wife a widow! 10 May his children wander about and beg, seeking food far from the ruins they inhabit! 11 May the creditor seize all that he has; may strangers plunder the fruits of his toil! 12 Let there be none to extend kindness to him, nor any to pity his fatherless children! 13 May his posterity be cut off; may his name be blotted out in the second generation! 14 May the iniquity of his fathers be remembered before the LORD, and let not the sin of his mother be blotted out! 15 Let them be before the LORD continually, that he may cut off the memory of them from the earth! 16 For he did not remember to show kindness, but pursued the poor and needy and the brokenhearted, to put them to death. 17 He loved to curse; let curses come upon him! He did not delight in blessing; may it be far from him! 18 He clothed himself with cursing as his coat; may it soak into his body like water, like oil into his bones! ... 20 May this be the reward of my accusers from the LORD, of those who speak evil against my life! [Psa 109:6-18, 20 ESV]

31 If the righteous is repaid on earth, how much more the wicked and the sinner! [Pro 11:31 ESV]
This seems like a good verse to pray.  A prayer of offense instead of defense.  Let the wicked be repaid on earth, now, today, in our time.

New Testament Considerations:
Are the verses below NT examples of these prayers?  First, note that prayers like this asking for God to take vengeance DO NOT apply to the direct violence of rulers, no matter what they may do.  There is no room here for that.
2 And the high priest Ananias commanded those who stood by him to strike him on the mouth. 3 Then Paul said to him, "God is going to strike you, you whitewashed wall! Are you sitting to judge me according to the law, and yet contrary to the law you order me to be struck?" 4 Those who stood by said, "Would you revile God's high priest?" 5 And Paul said, "I did not know, brothers, that he was the high priest, for it is written, 'You shall not speak evil of a ruler of your people.'" [Acts 23:2-5 ESV]  So...does this mean I cannot pray that corrupt rulers be struck down?  Or is it more that we cannot retaliate against a ruler for what he does to us directly?  Or was the problem that Paul said this publicly, though the place for it was in prayer?

9 And David says, "Let their table become a snare and a trap, a stumbling block and a retribution for them; 10 let their eyes be darkened so that they cannot see, and bend their backs forever." [Rom 11:9-10 ESV]
Paul is applying an Old Testament prayer of David to unrepentant Jews in Jesus' time.  This would surely seem to open the door for us to pray the same kind of prayers, asking God to destroy those among us who are violating His laws.

22 If anyone has no love for the Lord, let him be accursed. Our Lord, come! [1Co 16:22 ESV]
Not much doubt about this one.

The New Testament injunction that causes me to pause in all this is Jesus' own command that we love our enemies.  There are two similar verses, one in teh OT, in Proverbs, and the other in the NT:
21 If your enemy is hungry, give him bread to eat, and if he is thirsty, give him water to drink, 22 for you will heap burning coals on his head, and the LORD will reward you. [Pro 25:21-22 ESV]
And then:
44 But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you; [Mat 5:44 KJV]
The Proverb seems to be mostly about captured enemies.  It is written by King Solomon, about how to treat prisoners of war.  The advice is quite different than the prevailing practices of the time where prisoners were often tortured and killed.  Solomon is speaking of those taken captive in battles between nations.  He is not talking about individuals.  MSB note says that in those days, the way you melted metals for working with them was by covering them in hot burning coals.  That melted the hard metal.  Solomon is saying giving food and water to your enemies melts their hearts in that same way, and makes them more positively disposed toward you.  It takes the hatred from them, it changes their attitude toward you, it takes the fight out of them.

In Matthew, Jesus is not speaking as a King.  He isn't giving advice to rulers.  Jesus is speaking to individuals about how to lead their own lives.  What Jesus is relating is a strategy to win some by not responding in the "standard" way to abuse.  It is about love in return for hate in a way that clearly says the hatred is misplaced.  The contrast between the hate and the response to it is a strategy to make the hater pull up short, and consider, and maybe repent.  Solomon's proverb is similar in that treating enemies humanely may cause them to reconsider their mission.  Are they really fighting an enemy, or a friend who's been portrayed as an enemy.

On a more personal level, I don't believe any of these verses are about letting a violent looter into your house and offering him a cold drink because his looting and burning has him overheated.  I don't think it means this at all.  If so, Jesus wouldn't have said this:
43 But know this, that if the master of the house had known in what part of the night the thief was coming, he would have stayed awake and would not have let his house be broken into. [Mat 24:43 ESV].  We are certainly allowed to defend ourselves in our own homes.  This is not where we love our enemies.  Jesus does not include those who break in and threaten violence against your family in the "love them no matter what" category.  In Mt 5:44, above, He says those that curse you - that's words not physical violence, those that hate you - an attitude not an action, despitefully use you - put you in harms way, give you the lousiest assignment at work, cheat and rob you "legally" with no regard.  But this is not a physical thing.  Persecution.  Persistent, ongoing hounding, cheating, slandering, demeaning.  None of these imply one on one physical violence.  The thief at your house does imply violence, and I think this makes clear that the thief is not in view as an enemy to love.  At least not while he is acting to break in.

As I studied this, I found some differences in translation between the KJV and the ESV.  If that is of no interest, skip on down to the *** to continue the discussion.
Look at the difference in translation between the ESV and KJV:
44 But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you; [Mat 5:44 KJV]
44 But I say to you, Love your enemies and pray for those who persecute you, [Mat 5:44 ESV]
The newer translations certainly bring things down to the basics, making the KJV seem like some sort of amplified Bible.  I note that the interlinear translation in the online Blue Letter Bible is tied to the Textus Receptus.  When I switched from KJV to ESV and looked again at the interlinear, the word count is different and consequently the translation is "shorter".   The ESV translation is based on the morphological Greek New Testament, not on the Textus Receptus.  That is the root of the difference.

So the next question is, why does ESV use a different source than KJV.  The link below goes to a discussion of that very question on "Quora":
https://www.quora.com/What-is-the-morphological-Greek-New-Testament-MGNT?share=1

I don't know what Quora really is or who runs it or anything like that.  The explanation of each text given in Quora makes sense to me and would lead one to believe that perhaps the ESV is the better translation.  Indeed, the TR may be an amplification.  Since the mGNT is an attempt to look back through the lens of time to what the original actually said, one could accept it.  On the other hand, if one considers that the hand of God has preserved His Word as He wants it read through the generations, one might prefer the TR.

Looking just a little further, I see that the Textus Receptus dates from about 1500, and the mGNT is significantly more recent.  The "Aland" version especially, as Mr. Aland wasn't even born until 1915.  So there's that.

***Does it matter which translation I use to make my argument that Solomon and Jesus were speaking of different situations, but similar principles when they discussed how to treat our enemies?  No, I don't think so.  No matter which NT translation you use, the final purpose of treating your enemies as recommended is to change their minds about their captor or about us, for similar reasons.  In both cases, the goal is to de-escalate the bias and hatred so that communication is possible.  Solomon's goal is to melt hearts as burning coal melts metal.  To turn enemies into friends.  Jesus' command is to gain an open door to share the gospel with those who would abuse us.  

The purpose of loving your enemies in the NT is explicitly stated in the verse right after "love your enemies":
45 so that you may be sons of your Father who is in heaven. For he makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust. [Mat 5:45 ESV].  There is practically no difference in ESV and KJV on this verse.  We do as Jesus commands us in Mt 5:44 because that is how be become more like our heavenly Father.  He is patient, and longsuffering, and withholds His anger though many abuse His name.  We have already seen that the Proverb of Solomon is also about changing hearts.

The MSB note on the burning coals proverb also says that Paul quotes the verse in the NT:
19 Beloved, never avenge yourselves, but leave it to the wrath of God, for it is written, "Vengeance is mine, I will repay, says the Lord." 20 To the contrary, "if your enemy is hungry, feed him; if he is thirsty, give him something to drink; for by so doing you will heap burning coals on his head." [Rom 12:19-20 ESV]
So Paul says we are not to go off avenging ourselves on our enemies, but to treat them with kindness, aiming at changing their heart toward us, and leaving vengeance to God.  Leave justice also to God.  Paul is tying that Old Testament Proverb to the Sermon on the Mount.

My conclusion, therefore, and for today at least, is that prayers like David prayed are as acceptable, as relevant, and as powerful today as they were when David wrote his psalms.  Why indeed should it be a sin to pray that God would hurry and deal out the justice that He promises to deal out, and to do it not only in this world but in the world to come.  I end with this verse, which I believe is the synopsis of what these verses all say:
31 If the righteous is repaid on earth, how much more the wicked and the sinner! [Pro 11:31 ESV]

My prayer is that God would repay!