The furor over spousal abuse allegations that forced the resignation of one of President Trump’s top advisers last week has exposed fissures within the White House that had been papered over since John F. Kelly took over last summer as chief of staff with a mandate to end the dysfunction.

Aides to the president said they remained confused and upset over the handling of the accusations against Rob Porter, the staff secretary who stepped down. Days after his departure, the White House was still struggling on Sunday to provide a consistent explanation of who knew what and when, even as questions swirled about whether anyone might be felled as a result.

While said to be frustrated with Mr. Kelly over this and other issues, Mr. Trump sought this weekend to tamp down talk that he would be pushed out, instructing an aide scheduled to appear on television to report that the president still had confidence in his chief of staff. But other advisers to Mr. Trump maintained that his staff had let him down and that someone else should be held accountable.

“I spoke to the president last night,” Kellyanne Conway, a counselor to Mr. Trump, told Jake Tapper on Sunday on CNN’s “State of the Union.” “I told him I would be with you today. And he said, ‘Please tell Jake that I have full faith in Chief of Staff John Kelly and that I’m not actively searching for replacements.’ He said: ‘I saw that all over the news today. I have faith in him.’”