"Don’t kill them" (59:11), and two verses later, "Consume them" (59:13). Which is it? And how can a prayer ask for that?

First, what the verses say. "Don’t kill them, or my people may forget. Scatter them by your power, and bring them down, Lord our shield" (59:11) (tap forget). Two verses later: "Consume them in wrath. Consume them, and they will be no more. Let them know that God rules in Jacob, to the ends of the earth" (59:13). The first gives a reason, that God's people not forget; the second goes on, "Let them know that God rules in Jacob" (59:13). Consume is the verb kalah, twice, which Brown-Driver-Briggs gives in this form as "complete, bring to an end, finish". The psalm puts the two requests close together and doesn't say how they fit, and these notes don't choose a way to fit them. The requests against them are prayed to God (59:5, 59:11, 59:13); the psalmist doesn't say he will act against anyone himself. Psalm 59 is not the only psalm that prays against enemies, and Christians read these prayers in more than one way. The chapter on Psalm 109 sets out the main readings, each resting on its verses, and chooses none. These notes don't choose either.