This summer, the White House has faced a self-inflicted firestorm over its immigration policies, which led to thousands of family separations at the border; questions about why Mr. Trump stood by Scott Pruitt, the former administrator of the Environmental Protection Agency, who faced more than a dozen investigations into his spending and management practices before eventually resigning in July; and Mr. Trump’s cozy, widely criticized news conference with President Vladimir V. Putin of Russia in Helsinki, Finland.

On Wednesday, several aides dismissed the news about Mr. Cohen as just another bad headline lacking the silver bullet that they say the special counsel would need to prove that the president conspired with Russian officials.

Mr. Trump spent the early hours of Wednesday tweeting — he called the convicted Mr. Manafort a “brave man” who, unlike Mr. Cohen, “refused to ‘break’” or “make up stories in order to get a ‘deal.’’’ He also monitored headlines, as he did after his news conference with Mr. Putin. In the interview with Fox News, he asserted that money for the payments to the women had come not from his campaign, but from his own accounts.

“I don’t know if you know,” Mr. Trump said during the interview, “but I tweeted about the payments. But they didn’t come out of the campaign.” Campaign finance laws still prohibit Mr. Trump from making unreported payments related to the campaign, regardless of where they came from. Neither payment was disclosed to the Federal Election Commission.

On Air Force One on Tuesday night on the way back from a rally in West Virginia, Mr. Trump repeatedly minimized the news, telling aides that the legal developments were not about him, but about Mr. Manafort and Mr. Cohen. He also groused over the optics of the rally, telling a person close to him that the crowd seemed flat and that some chairs were empty.

By Wednesday, Mr. Trump’s lawyers were arguing privately that Mr. Cohen’s admission and guilty plea to violating campaign finance laws was a punch but not a knockout blow, and were assessing what options they had for fighting back. They stressed that Mr. Cohen had said repeatedly in previous accounts that Mr. Trump was not aware of his payment to Ms. Clifford, known as Stormy Daniels.