“When I came into this business, it had been going on for many years,” Mr. Cohen said.

“It’s an opportunity,” Mr. Cohen said of the event. “If I go to the prayer breakfast, I have a good chance of maybe shaking the president’s hand or talking to him for two minutes.”

“In a way, it bypasses protocols,” he added, “but in a way, it is taking advantage of people being present in the same venue.” Such invitations to foreign leaders, he said, are “very useful to them back home.”

James C. Slattery, a former Democratic congressman from Kansas whose firm was paid more than $1.8 million since 2011 to provide lobbying help for Ms. Tymoshenko, the Ukrainian leader, and her political causes, encouraged her to attend the breakfast. He accompanied her to last year’s event, where she secured a photograph and brief conversation with Mr. Trump, which her allies used to promote her nascent presidential candidacy back home — to the apparent surprise of the White House, and to the chagrin of the sitting Ukrainian president.

Held every year at the Washington Hilton, the prayer breakfast festivities span several days during the first week of February, with the American president appearing at a ceremonial breakfast on Thursday. The days are packed with programming, after which guests head to private suites with names like the Africa room and the Middle East room, or to fancier hotels in nearby Georgetown, where they mingle late into the night — praying, sharing business cards and sometimes draining expensive bottles of cognac. A favorite activity is an annual midnight tour of the Capitol, hosted by a former congressman.

Some describe the gathering as similar to the World Economic Forum, except that Jesus is the organizing principle. The eclectic guest list has included the Dalai Lama, the Rev. Billy Graham, Mother Teresa, the singer Bono and the former Redskins coach Joe Gibbs, as well as the Palestinian leader Yasir Arafat and President Paul Kagame of Rwanda.