Parks officials said that under the terms of the contract, the clubhouse must be completed by March 31, 2020. They added that the Trump Organization “has generally maintained the golf course in a satisfactory manner.”

Mayor Bill de Blasio, who has been a vocal critic of Mr. Trump, has previously called for a legal review of all city contracts with connections to Mr. Trump, including the one for the golf course.

“As long as they’re upholding the contract terms, which were negotiated by the previous administration, we can’t cancel this contract,” said Natalie Grybauskas, a spokeswoman for the mayor. “Our focus is on improving the surrounding parkland, including developing a waterfront path next to the course to serve residents who live nearby.”

Mr. Lieberman said that the golf course had already benefited the Bronx and the city. Since 2015, the golf course has hired about 400 workers, most New Yorkers. In addition, he said, it has supported youth programs, patronized local bakeries and breweries, donated golf carts to a local veterans hospital and offered discounted fees to veterans and older people.

Joe DeSiena, 35, a teacher at Abraman Lincoln High School in Brooklyn, said the course was expensive but unlike anything he had experienced in the five boroughs. “This is country club quality and a public course,” he said.

At the course this week, a tent was being used as a temporary clubhouse and a wooden building was the pro shop. One caddie, who refused to provide his name for fear of losing his job, said that as a Democrat who did not support Mr. Trump, it was difficult working at the course. His friends gave him a hard time, too.

In the neighborhood, many Bronx residents said they had never tried the golf course because it was too expensive. The average household income for the Bronx is $51,513 annually, compared with $89,680 citywide, according to an analysis of census data by Queens College. The fee for playing 18 holes on the Trump course over the weekend is $175 for a city resident and $227 for a nonresident, the highest of any of New York’s municipal courses.