But the agreement hashed out by Mr. Pence in Ankara did please a familiar audience of one.

“We’re doing very, very well with Turkey,” the president reiterated to reporters in the Oval Office on Friday. “There’s a cease-fire, or a pause, or whatever you want to call it.”

One possible clue that an association with the agreement might not be a winner, especially for a Republican, came from Secretary of State Mike Pompeo, who arrived in Ankara on a separate plane from Mr. Pence, and made it clear he was there strictly in a support role to back up the vice president. (His trip also extended to Jerusalem and Brussels, and his plane departed before Mr. Pence’s.)

“If the issue is how Trump sees it, Pence is going to come out looking great,” said Eric S. Edelman, a former ambassador to Turkey and a senior Defense Department official in the George W. Bush administration. “If it’s from the point of view of Republicans who are very worried about this, does Pence come out looking great? No. What will it look like in historical terms? It’s too early to tell.”

There was a sense of foreboding about Mr. Pence’s task among administration officials traveling with him on the flight to Turkey, some of who noted with regret that the vice president was forced to reschedule events related to the United States-Mexico-Canada Agreement in Wisconsin for the trip.

Senior administration officials had expressed reservations about Mr. Pence going at all, arguing that sending him to meet with Mr. Erdogan was something of a reward to the Turkish president in and of itself, a chit that was better held for when more progress had been made on a deal.