Some Democrats from pro-Trump states said they had no immediate plans to highlight the matter of the president authorizing an undisclosed payoff to a pornographic film actress with whom he had an affair.

“If asked, I absolutely will talk about it,” said Senator Jon Tester of Montana, adding that it is not currently a front and center campaign issue. “I think people are still trying to sort through it.”

Given the health of the economy and relative stability abroad, Mr. Trump’s approval ratings are already significantly lower than what virtually any other president would be enjoying at this moment of his administration. And if there is one predictable element of this otherwise unpredictable presidency, it is that some new story will detonate in the days or weeks ahead, pushing the last eruption off the home page and television chyron.

Indeed, Republicans are wishing that the Manafort and Cohen felonies only wind up being this week’s version of the Omarosa tapes — the latest ephemeral drama to captivate the capital but barely faze voters. But party officials are apprehensive about what more Mr. Cohen may reveal on his way to a likely prison sentence.

Even more worrisome to Republicans is what damage Mr. Trump may do to himself. If the president is seen as thwarting Mr. Mueller’s investigation, either by terminating the special counsel or pardoning allies who are implicated, it would create a far more serious upheaval and force Republican lawmakers into a confrontation they have long avoided.

“Of course you can imagine the president doing things that would be counterproductive,” said Senator Bill Cassidy, Republican of Louisiana, before adding with evident hope in his voice, “but you also could imagine the president saying something but never acting on it, just venting.”