SEOUL — Look for a fourth summit between President Trump and North Korea’s leader, Kim Jong-un, this time against the background of the North’s new alliance with Russia and the war in Ukraine.

Whenever they do meet, Messrs. Trump and Kim should have much to talk about in addition to the North’s nuclear program, which Mr. Kim refuses to abandon. Mr. Kim’s enhanced alliance with President Putin adds a whole new dimension to the North-South Korean standoff, as well as to the war in Ukraine and America’s relations with Moscow.

For one, reports of North Korea’s plans to send fresh troops to fight for the Russians complicate efforts to bring the two leaders together while Russia provides the North with the technology for building more and “better” nuclear warheads and long-range missiles.

With 1.2 million troops in his armed forces, Mr. Kim may not be too concerned about heavy casualties inflicted on the 11,000-12,000 men he’s already sent to Ukraine. South Korea’s joint chiefs of staff has stated in writing that North Korea “is assumed to be accelerating preparations for additional measures and deployment amid multiple casualties,” according to Seoul’s Yonhap News. Moreover, “Preparations for a spy satellite or an ICBM seem to be continuing.”

The prospect of another Trump-Kim meeting arose when Fox News’s Sean Hannity asked Mr. Trump if he would like to see the North Korean leader again, and Mr. Trump responded, “Yeah, he liked me, and I got along with him.”

For Mr. Kim, the rewards for dealing with Mr. Putin would seem likely to surpass anything Mr. Trump may have to offer — that is, unless Mr. Trump is willing to offer far more.

In the Ukrainian capital, Kyiv, a website and YouTube channel, RBC-Ukraine, reports a Ukrainian political strategist, Zaras Zahorodrill, suggesting that Mr. Trump “could potentially compel North Korea to cease its support for Russia.”

“He could say: ‘How much is Putin giving you? I’ll give you more,’” Mr. Zahorodrill is quoted as saying. Alternatively, he says, Mr. Trump might tell Mr. Kim: “You do realize that American strategic bombers also have targets in North Korea — don’t play games, this will end very badly for you.”

Mr. Trump, though, is not likely to engage in such threats while looking for another seance with Mr. Kim, with whom he professed he “fell in love” in previous meetings. On the way to another summit, he’s already upset America’s South Korean ally by remarking that Mr. Kim “is a nuclear power” — a comment that went against the longstanding American policy of not recognizing North Korea as a member of the global nuclear club.

Embroiled in domestic political controversy, South Koreans worry that Mr. Trump will overlook their concerns in his eagerness to see Mr. Kim. South Korea’s foreign minister, Cho Tae-yul, placed a call to the newly appointed American secretary of state, Marco Rubio, who assured him of “close cooperation on North Korea’s nuclear issues,” according to Yonhap.

Mr. Trump, Koreans complain, was in effect admitting he had gotten nowhere in his first summit with Mr. Kim in Singapore in June 2018. His second summit with Mr. Kim, at Hanoi in February 2019, broke up when Mr. Kim refused to give up his nuclear program. Their third meeting, at the truce village of Panmunjom the following June, ended inconclusively.

Since the excitement of those meetings, the North Korean menace has only gotten worse. Mr. Kim, reassured by his alliance with Russia, has declared the South “the enemy” while providing the Russians not only with troops but also with artillery shells and other weaponry to wage war in Ukraine.

To Koreans, Mr. Trump showed his lack of understanding of the depth of the issues when he asked American troops in Korea, seen on a projector screen shown at his inaugural ball at the White House, “How are we doing over there? How’s Kim Jong-un doing?”

The troops laughed at the question, which they had to assume was in jest. Mr. Trump, though, will be looking for the real answer if Mr. Kim, from his new position of strength, agrees to see him.