WASHINGTON — President Trump’s mood went from feisty to self-pitying to deflated on Wednesday as he fended off questions about a July phone call in which he urged the president of Ukraine to work with Attorney General William P. Barr on potential corruption investigations connected to former Vice President Joseph R. Biden Jr., a Democratic rival.

Although Mr. Trump sought to present a business-as-usual image in his annual trip to New York for the United Nations General Assembly — by highlighting a trade deal with Japan, among other things — his anger and anxiousness took over his day, aides said. He appeared aggrieved in comments to reporters ahead of a meeting with the Ukrainian president, Volodymyr Zelensky, but by late afternoon, Mr. Trump appeared so exhausted that he spoke unusually slowly at a rambling news conference meant to sum up his trip at the United Nations.

He opened by naming 28 countries whose leaders he had met with and said that unfortunately reporters were far more interested in his phone call with Mr. Zelensky. “I’ve been up from early in the morning until late in the evening and meeting with different countries, all for the good of our country.,” Mr. Trump said. “And the press doesn’t even cover all of this.”

Then he moved on to the call, and the words he said spoke to the Ukrainian president.

“They were perfect,” Mr. Trump said. “They were all perfect.”

He added: “I didn’t threaten anybody. In fact, the press was asking questions of the president of Ukraine and he said, no pressure. I used the word pressure and I think he used the word push, but he meant pressure, but it’s the same thing. No push, no pressure, no nothing. It is all a hoax, folks. It is all a big hoax.”

The president went so far as to suggest that his enemies had intentionally sabotaged his excellent adventure in Manhattan. “So that was all planned like everything else,” Mr. Trump said. “It was all planned. And the witch hunt continues,” he said.

Aides who had lived through the day-to-day nerves of the investigation by the special counsel, Robert S. Mueller III, for roughly two years had thought the worst was behind them. But they said the growing storm around their boss’s call with Mr. Zelensky drove home the new reality that an impeachment inquiry would consume the next months.

Although several of the president’s allies sought to play down the contents of the reconstructed transcript of the call released Wednesday, two people close to Mr. Trump said that the transcript matched what they knew of his dealings with world leaders on the phone. One former senior official called it the typical playbook: Engage in flattery, discuss mutual cooperation and bring up a favor that then could be delegated to another person on Mr. Trump’s team.

At the White House, a grim sense of frustration has set in. Aides said they thought Democrats had gone too far. Nonetheless, they acknowledged that the impeachment inquiry represented the likely death of any hope of passing legislation in the next 13 months.

Several expressed fear that other witnesses would come forward in relation to Mr. Trump’s contacts with the Ukrainian president, or that other whistle-blowers on other matters would emerge.

The White House counsel’s office is prepared for an impeachment inquiry, but other departments in the West Wing are badly depleted by staff departures and plagued by exhaustion.

In the morning, White House officials briefed about a dozen Republican members of the House and Senate on the reconstructed call, providing them with talking points. Mr. Trump called into the meeting as the members sat around the table in the Roosevelt Room, insisting it was a hoax and that Democrats had gone overboard in pursuing him, people briefed on the meeting said.

Some of Mr. Trump’s allies said he saw impeachment as a good political opportunity that would result in a backlash against the Democrats. At the same time, several people close to Mr. Trump said he did not want to be the third president in American history, after Andrew Johnson and Bill Clinton, to be impeached.

Some Trump allies, including Newt Gingrich, who served as the Republican House speaker during the Clinton impeachment, saw an upside.

Mr. Gingrich, an occasional Trump adviser, said in an interview that an impeachment inquiry would lead to further questioning into the Biden family’s financial and political activities abroad. “Nancy Pelosi ended Joe Biden’s campaign, and she doesn’t even realize it,” Mr. Gingrich said in an interview. “The Democrats will run further into a cul-de-sac chanting ‘impeachment’ as a policy.” Of the Clinton impeachment, he said that “it was to our disadvantage politically,” adding that regular voters grew frustrated with a focus on impeachment over matters like the economy.

Anger from some current and former administration officials was directed at Rudolph W. Giuliani, Mr. Trump’s longtime friend and personal lawyer, who had repeatedly told Mr. Trump that he believed Ukraine was involved in the 2016 election interference, a notion on which the president fixated. The aides believe that Mr. Giuliani, eager to land something that would please Mr. Trump, had gotten too far out on a limb. And they cringed at his television appearances.

Mr. Trump had little chance to catch up with his media coverage at the United Nations, but aides said they were bracing for the president to react angrily when he finally saw some of it after a fund-raiser on Wednesday night in New York, to be held at the Upper East Side home of John Paulson, a hedge fund manager.

Mr. Trump, as he prepared to leave the news conference for a bit of downtime before his evening fund-raiser, vented his frustrations one last time. He complained that his diplomatic work here had largely been ignored.

“Unfortunately the press doesn’t even cover it,” he said. “We have a tremendous trade deal with Japan that doesn’t even get covered because you waste your time on nonsense.”

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