What delicate issues are at the very top of Claudius's agenda for the meeting of his court in Hamlet?

Just to clarify some things:

By the time of the speech before the court (Act 1 scene 2), Claudius and Gertrude are already married. Indeed, Claudius generally thanks those gathered for previously offering their advice about the marriage.

The next bit of business is of note in that it involves young Fortinbras, the son of the elder Forintbras who was killed by Prince Hamlet's father. It also involves an uncle, the elder Fortinbra's brother who is sick and knows little about young Fortinbra's plan to reclaim land lost by Norway to the dead King Hamlet. Claudius says that he plans to send Cornelius and Voltimand to get the old uncle to stop young Fortinbras in his foolish venture.

Next, the King brings up the subject of Laertes who wants to go back to school. After asking Laertes' father, Polonius, if it's OK with him, Claudius grants Laertes his wish.

The same wish, to go back to school, is not granted to Hamlet. In this last bit of business before the court, the King is adamant: Hamlet is to stay for a while in Denmark. The King wants to keep an eye on Hamlet, but he certainly does not share this reason, nor his wariness, with the rest of those assembled.