What political problems does Claudius now face as a result of Polonius’s death?
The only real political problem Claudius faces is that he is now without an advisor and a confidant, a person who would spy for him and keep tabs on Hamlet and inform him about his actions so that he may counter whatever Hamlet may be scheming.
Furthermore, it is clear that Polonius was in charge of palace-affairs, supervising and directing, making arrangements and keeping the King informed about the comings and goings of guests and other visitors - a sort of 'charge-de-affaires'. Claudius will have to appoint a replacement for Polonius which would obviously be a difficult task, since the two of them had had a long-standing relationship and had developed trust.
Polonius had also been very dutiful and essentially servile, as he himself tells Claudius, in Act 2, scene 2:
I assure my good liege,
I hold my duty, as I hold my soul,
Both to my God and to my gracious king:
Claudius thinks of him as 'a man faithful and honourable.'
A replacement would, in all probability, not display the same obedience that he had. It would now be much more difficult for Claudius to keep an eye on Hamlet and counter the threat that he obviously holds. Also, as far as trust goes, it would be difficult to determine where the new appointee's true allegiance really lies. With Hamlet or with him? Hamlet was quite popular and it would thus be difficult to find a trustworthy replacement for Polonius.
In this sense, then, Claudius would have to carefully consider his choice.
This is, however, not Claudius' only challenge. He would obviously have to deal with Laertes' anger at the murder of his father. Laertes threatens:
To this point I stand,
That both the worlds I give to negligence,
Let come what comes; only I'll be revenged
Most thoroughly for my father.
He also needs to address Ophelia's grief. He does the first with aplomb, and manages to turn Laertes' anger to Hamlet, going even further by plotting Hamlet's assassination by Laertes' hand in a pernicious plan involving a duel.
As far as Ophelia is concerned, there is not much he can do but express his concern. She becomes so overwrought that she literally commits suicide (or, more accurately, allows herself to die by drowning).