Tucker Carlson can probably nullify every one of those contract provisions by announcing that he’s starting a political action committee, or even perhaps announcing that he’s a candidate for public office.
Carlson has to negotiate an end of the contract, or cork himself until it expires at the end of next year.
Fox has, in hand, a commitment on Carlson’s part to remain effectively silent on all topics relating to what Fox hired him to opine on. Which was pretty much everything.
He can speak in generalities but will want to avoid specifics so as not to give Fox a predicate on which to base a lawsuit.
Fox will always have a right to sue if they conclude based on competent legal advice that Carlson has acted in such a way as to violate his employment contract and thereby give them sufficient substantive grounds for suit.