A few of the New Yorkers are transplants: Ms. Powell was born in Cairo and moved to Texas and Washington before ending up in Manhattan at Goldman Sachs. Mr. Cohn, who rose to the No. 2 job at Goldman Sachs, is originally from Shaker Heights, Ohio. But even the transplants sometimes behave as if they, and their boss, never left the five boroughs.

When Mr. Cohn, the president’s chief economic adviser, briefed reporters at a summit meeting in Sicily in May, he boasted that Mr. Trump had been working nonstop since leaving New York a few days earlier.

“Washington,” the national security adviser, Lt. Gen. H. R. McMaster, leaned in to correct him.

Presidents often bring a retinue from their home states that gives the White House a distinct cultural accent — often to the disdain of Washington’s permanent establishment. Jimmy Carter’s young aides were quickly labeled the “Georgia Mafia.” Richard M. Nixon and Ronald Reagan each came with a powerful, not always popular, California contingent.

The closest analogy to Mr. Trump’s White House was Franklin D. Roosevelt’s. A former governor of New York, Roosevelt populated his administration with Columbia University professors and influential New Yorkers like Henry Morgenthau Jr., who was his Treasury secretary.

“How much it changes things depends a great deal on how experienced they are with Washington politics,” said Robert Dallek, the presidential historian, who will publish a new biography of Roosevelt this fall. “There is a Washington culture that often surpasses what they may have brought with them from Georgia or Texas or New York.”

As much as Mr. Trump’s New York contingent has set the tone in the West Wing, its influence on the administration’s policy has been, at best, mixed. Mr. Cohn, along with Mr. Kushner and Ms. Trump, failed to talk the president out of withdrawing from the Paris climate accord. They have not appreciably softened his policies on trade or immigration.

Mr. Trump’s speeches and policies still bear the unmistakable imprint of his senior policy adviser, Stephen Miller, and his chief strategist, Stephen K. Bannon. Both are ardent nationalists who are scathing about the mainstream tendencies of the New York crowd — and neither is a New Yorker. Mr. Miller grew up in Santa Monica, Calif.; Mr. Bannon was born in Norfolk, Va., though he lived in New York when he worked for Goldman Sachs.