WASHINGTON — On the day after he made more news than any chief of staff in recent White House history — much less an acting one —Mick Mulvaney went about his business as usual.

He finalized plans for hosting Republican members of Congress this weekend at Camp David, a form of outreach he has regularly pursued and a bit of traditional Washington socializing that is not President Trump’s forte. He was booked as a guest on Fox News’s Sunday morning talk show, despite a desire by some conservatives that he stop talking. And he gave a speech to a Republican group in the Raleigh, N.C., area.

But Mr. Mulvaney’s job has been anything but normal since the news conference on Thursday at which he seemingly undermined the Trump administration’s strategy for avoiding impeachment by acknowledging that Mr. Trump had sought a quid pro quo for providing Ukraine with American aid. In the chaotic aftermath, the president’s Republican allies are questioning Mr. Mulvaney’s savvy and intelligence even as the Trump campaign is defiantly turning one of his lines from the news conference into a T-shirt.

As he approaches his anniversary in the White House, Mr. Mulvaney, 52, a former South Carolina congressman and Trump budget director, finds himself in a strange netherworld.