What argument does Claudius give for not persecuting Hamlet in Shakespeare's Hamlet?
I am not sure if I totally understand your question. Claudius does plot to get rid of Hamlet behind his back, trying first to use Rosencrantz and Guildenstern to have him executed in England, and then poisoning Laertes' dagger as well as the wine to have a backup plan to kill Hamlet.
Might you be referring to Claudius' argument for not persecuting Hamlet over his killing of Polonius? Laertes believed that Claudius had some part in Polonius' death, and Claudius explains to Laertes that the reason he did not pursue an inquiry into Polonius' death is because Gertrude, Hamlet's mother, loves him very much, and also Hamlet is extremely popular with the people. He swears to Laertes that he had nothing to do with killing Polonius, and hence Laertes challenges Hamlet to a duel, where Claudius is hoping his poison will do the trick, one way or the other. Of course, his plan backfires.
Claudius says two things: that people of Denmark love Hamlet and think of his as fair and that his mother adores him. Those are the two reasons that Claudius gives for not prosecuting Hamlet. It is very different from what he believes, though, as he has an alternative plan for Hamlet.