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This is pretty broad topic. This ranges from paying a person to not revealing (non-criminal) information they know to selling exclusive rights to their life story.
Should elected officials be allowed to have private contracts that prevent an individual from revealing information, even if it is paid for out of their own money?
If you're talking about Trump here, (adulterer in chief) some of these escapades happened before he became president, some of these came to light during his campaign, see, people should have know better than to vote for some man who habitually abused women at every turn, but, people loved this jerk, even women voted for him because they thought he was going to be such a great president and was going to make things great in America.
If someone paid me to keep quiet about a relationship and if that person were a politician or running for public office, I believe the general public would need to know what kind of a jerk they're voting for, so, no, a gag order as you call it should not apply, but we'll let the lawyers sort it out in court.