UNITED NATIONS — Just after 8 a.m. Monday, President Donald Trump set off from Trump Tower for his annual week on the diplomatic stage at the U.N. General Assembly. By the time he walked into the General Assembly building 10 minutes later for a session on the illicit global drug trade, domestic politics had already intervened.

Reporters waiting there asked him about the sexual misconduct allegations against his Supreme Court nominee, Judge Brett M. Kavanaugh, which had been published the night before in The New Yorker. Trump said the tangle of charges brought first by Christine Blasey Ford and now Deborah Ramirez could be one of the “single most unfair, unjust things to happen to a candidate for anything.”

Seizing the opportunity to defend his nominee, Trump said he stood behind Kavanaugh and dismissed the claims of the two women as “totally political.” Then he walked on into the chamber and spoke to the assembled members about the evils of drug trafficking around the world.

Political tempests often intrude on presidents, whether they are traveling overseas or playing host to foreign leaders at the White House. Still, it is hard to recall a General Assembly that opened against a backdrop of such political turmoil, on so many fronts.

As Trump performed the rituals of a statesman — shaking hands with allies, signing a free-trade agreement with South Korea — he kept getting drawn back into the political storms in Washington. At one point, he tried to regain the initiative, announcing that he planned to meet North Korea’s leader, Kim Jong Un, in the “not-too-distant future.”

In normal times, such an announcement would have commanded major headlines. But it was largely lost in a second drama that erupted Monday over the job status of the deputy attorney general, Rod J. Rosenstein.

Shortly after 10:30, reports began circulating that Rosenstein was on his way to the White House, where he expected to be dismissed. His fate had been unclear since The New York Times reported last week that he offered to tape his conversations with the president and discussed invoking the 25th Amendment to remove him from office.

White House officials had scheduled a briefing for 11 a.m. with Secretary of State Mike Pompeo; the U.S. ambassador to the United Nations, Nikki R. Haley; and the national security adviser, John R. Bolton. And the three officials showed up determined to press the administration’s foreign policy priorities, Iran and North Korea.

“Happy UNGA!” a cheery Haley said, using the clunky acronym for the United Nations General Assembly. “We got a great start today.”

By then, however, reporters wanted to know only whether Haley and Rosenstein had ever discussed removing Trump from office. Both she and Pompeo indignantly denied taking part in, or hearing, any conversations about the 25th Amendment. They also took the opportunity to pledge their fealty to the president.

“I have been at the center of this administration, along with lots of other folks, from virtually Day 1,” Pompeo declared. “I’ve never heard anyone talk about it, whisper about it, joke about it in any way.”

Haley said: “It is completely and totally absurd. No one is questioning the president at all. If anything, we’re trying to keep up the pace with him.”

As it happens, Trump had some down time Monday to dwell on Rosenstein. After his morning meeting on drug trafficking, he returned for several hours to Trump Tower, where he and Rosenstein had a phone conversation, at Rosenstein’s request. The two men agreed to defer any action for a couple of days.

“We’re going to have a meeting on Thursday,” Trump told reporters later, as he sat down with President Moon Jae-in of South Korea. “We want to have transparency, we want to have openness, and I look forward to meeting with Rod at that time.”

“Right now, today, we’re meeting with a lot of great people, including President Moon,” he said, gesturing to his smiling guest.

Administration officials said working-level diplomacy at the United Nations was largely unaffected by the drama surrounding Rosenstein and Kavanaugh. But one official said it did preoccupy Trump’s top advisers, noting that it was a big topic of conversation over the weekend at the president’s golf club in Bedminster, New Jersey.

The fact that the secretary of state and the ambassador to the United Nations had to deny taking part in a plot to remove the president, in a week when they are meeting with dozens of foreign officials, illustrated the unique challenges of serving this president. Even without the domestic distractions, Trump’s aides faced a hectic week of diplomacy. On Monday morning, Bolton warned Russia not to send powerful air-defense missiles to the Syrian government of President Bashar Assad. Russian officials said they planned to send the S-300 missiles to Syria after a Russian plane was mistakenly shot down last week — an act that Moscow blamed on Israeli pilots.

Trump also continued to project notably more optimism than his staff about the prospects for a nuclear deal with North Korea. He praised Kim for his “tremendous enthusiasm” for an agreement, and said they both looked forward to meeting again one on one, at a location still to be determined.

“We’ve made more progress than anyone’s made — ever,” Trump declared. “The relationship is very good. In fact, in some ways, it’s extraordinary.”

Pompeo, who is responsible for hashing out the terms of a nuclear agreement with Pyongyang, struggled to square his boss’ enthusiasm with the administration’s more pessimistic appraisal. “There remains work to be done,” he conceded. “There will be some time before we get to complete denuclearization, for sure.”

As the day wound down in New York, and attention in Washington shifted to a Fox interview with Kavanaugh and his wife, Ashley, Trump joined Moon again to sign the new trade agreement between the United States and South Korea. The president spoke of how many more cars U.S. automakers could sell in the Korean market; Moon said he was glad the two countries had settled their differences.

“I’ve never seen my name in Korean,” Trump joked as he signed the leather-bound document. “Looks nice.”

Then, as he and Moon shook hands and stood up to leave, there was one more question about Kavanaugh: Was the president still confident the Senate will confirm him?

“Well, I hope he’s going to be confirmed,” Trump replied, ending his day at the United Nations much as he began it.