Trump used his meetings with heads of state to flay Biden, celebrate his personal attorney’s altercations and tend to his media feuds. For instance, he argued during a sit-down with the Polish president that if Biden were a Republican, the news media would have him in “the electric chair by right now.”

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Trump’s head-turning visit to the United Nations came as scrutiny of his conversation with Zelensky intensifies in Washington. House Democrats demanded to see a transcript of the call as well as the intelligence officer’s whistleblower complaint that brought it to light, and argued that Trump’s alleged conduct represents an abuse of power and could be grounds for impeachment.

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The Ukraine episode hung over Trump’s first day at the annual U.N. meeting, beginning with a volley of shouted questions from journalists as he walked into the towering U.N. headquarters building in New York. He again admitted that he raised the Biden allegations in his call with Zelensky.

“It’s very important to talk about corruption,” he told reporters. “If you don’t talk about corruption, why would you give money to a country that you think is corrupt?”

Yet Trump struck an unapologetic note, barreling past accusations that he yet again improperly engaged with a foreign country. Echoing language he used to dismiss the Russia investigation led by Robert S. Mueller III, the former special counsel, Trump called the Ukraine development another “Democratic witch hunt,” and said any problematic dealings with Ukraine were done by Biden.

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“The one who’s got the problem is Biden,” Trump said.

Trump denied that he explicitly withheld $250 million in U.S. military aid to Ukraine unless its government launched a probe into the Biden family. He insisted his and Zelensky’s July 25 conversation had been “a perfect phone call,” and he hinted at releasing a transcript or notes.

“I hope you see it, frankly, and you will find out that I did not do that at all and you’ll be very disappointed when you see it,” Trump told reporters.

Despite the peril the Ukrainian episode presents for his presidency, Trump seemed to revel in the opportunity to kick up dust around Biden, who has led Trump in virtually every public poll for several months.

Trump is attempting to sully one of Biden’s calling cards in the 2020 campaign: His pledge to honor the rule of law and to restore moral integrity and ethics to the Oval Office. Trump is trying to portray Biden as corrupt and deny him this signature issue.

Within Trump’s orbit, some argue that the Ukraine news cycle will ultimately benefit the president — if he and his allies can redirect the public’s attention to what they claim is unethical behavior by Biden and his son Hunter.

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“We know that Hunter Biden had a sweetheart gig in Ukraine and that Joe Biden has bragged about trying to fire Ukraine’s top prosecutor looking into corruption,” Tim Murtaugh, a Trump campaign spokesman, said in a written statement.

Hunter Biden served for nearly five years on the board of Burisma, Ukraine’s largest private gas company, whose owner came under scrutiny by Ukrainian prosecutors for possible abuse of power and unlawful enrichment. Hunter Biden was not accused of any wrongdoing in the investigation. As vice president, Joe Biden pressured Ukraine to fire the top prosecutor, Viktor Shokin, who Biden and other Western officials said was not sufficiently pursuing corruption cases. At the time, the investigation into Burisma was dormant, according to former Ukrainian and U.S. officials.

In private talking points that the Republican National Committee sent out to surrogates Monday morning, the committee reiterates Trump’s claim that his conversations with Zelensky were “totally appropriate” — and then offers a half-dozen unflattering claims about Biden and Ukraine.

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In the run-up to the U.N. General Assembly, Trump’s open door to an unprecedented meeting with Iranian President Hassan Rouhani had been the most anticipated potential development.

That possibility now seems remote, and has taken a back seat to the Ukraine matter. Anticipation has also been building for Trump’s scheduled meeting Wednesday with Zelensky, who is making his largest international debut here since winning election in May.

“We don’t have, obviously, the clarity that we need regarding this meeting, but this is, I think, going to be the most-watched bilateral meeting at the General Assembly,” said Heather A. Conley, head of the Europe program at the Center for Strategic and International Studies.

At the United Nations, visiting diplomats were baffled by the sudden interjection of a complex Trump scandal that they found mystifying.

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“It puts foreign leaders in a difficult if not impossible position,” said Richard N. Haass, president of the Council on Foreign Relations. “How do you manage this complicated person who doesn’t play the game by the rules, and how do you somehow protect this important relationship? Everybody’s a bit wary.”

Victoria Holt, managing director at the Henry L. Stimson Center, said the arcane details of the Ukraine story and its overlap with U.S. politics are largely beside the point for other leaders here. “I’m sure they are scratching their heads,” she said. “The United States is supposed to be leading on major issues” such as climate change.

Trump stayed away from the Ukraine topic in his only major prepared remarks of the day, before a U.S.-led summit on “religious freedom” that also contained no reference to Trump’s proposed ban on Muslim immigration to the United States.

But the president tweeted about Ukraine and Biden on his way into that event, calling those who question his conduct “stone cold Crooked” and questioning the patriotism of the “so-called whistleblower.’”

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Trump kept up the theme when he met with Polish President Andrzej Duda. With Duda watching, Trump branded as “a faker” a journalist who asked him to elaborate on what he had discussed with Zelensky.

“Joe Biden and his son are corrupt, all right?” Trump added. “But the fake news doesn’t want to report it because they’re Democrats. If that ever happened — if a Republican ever did what Joe Biden did, if a Republican ever said what Joe Biden said, they’d be getting the electric chair by right now. Look at the double standards. You people ought to be ashamed of yourself. . . . You’re crooked as hell.”

Later, during a meeting with Prime Minister Lee Hsien Loong of Singapore, Trump praised last week’s combative television appearance by his personal attorney, Rudolph W. Giuliani, which some critics mocked as belligerent.

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“I watched Rudy take apart Fredo,” Trump said, referring to CNN anchor Chris Cuomo using a nickname from “The Godfather” movies. “Fredo’s performance was incompetent. . . . Rudy Giuliani took Fredo to the cleaners. First time I’ve watched CNN in a long time.”

The prime minister sat impassively at Trump’s side, perhaps bewildered by the American film reference.