Are the reasons Claudius gives Laertes for not punishing Hamlet for killing Polonius genuine?
No, Claudius's reasons for not punishing Hamlet for Polonius's murder are not genuine. Claudius is the king, and as such it is his duty to administer justice despite family obligation. If he truly wanted to charge and punish Hamlet he would have the power to do so. Rather than following the proper process for obtaining justice, Claudius manipulates Laeteres into plotting against Hamlet, all in the name of justice. The truth of that matter is that Claudius knows that Hamlet poses a threat to him and his rule. Hamlet clearly suspects foul play, which is illustrated by the play that he performs. If Claudius brings charges against Hamlet in the proper way, Hamlet will be given the opportunity to tell others about Claudius murdering King Hamlet. Claudius is not interested in getting justice for Polonius's death, he wants to keep his throne.