“There was probably a time when it made sense to have someone like Kevin McCarthy, but we need new leadership in the House,” said Joe Kent, a square-jawed former Special Forces officer who is trying to unseat Representative Jaime Herrera Beutler in Washington, one of 10 Republicans who voted to impeach Mr. Trump after the Capitol attack. “He’s used to a different era.”

He added, “Our job is to obstruct and impeach, not to cut any deals.”

One Republican House member who backs Mr. McCarthy, who insisted on anonymity to discuss his predicament candidly without fear of a backlash from colleagues or constituents, said that as hard as Mr. McCarthy was working to maintain control, some in the party were so extreme that his position had become all but untenable…

Current and former aides to Mr. Trump describe Mr. McCarthy’s relationship with the former president as cordial but lacking in any loyalty. They are not in lock step on which candidates to support in the midterm elections, and Mr. McCarthy knows he ultimately has limited influence over Mr. Trump’s endorsements. That has not stopped the House leader from trying.