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Months after landing its first job in nearly a decade on the New York City subway, a Long Island construction firm hosted an intimate fund-raiser in its luxury suite at Citi Field for a special guest: Gov. Andrew M. Cuomo.

The firm, Haugland Group, had wanted to expand into the lucrative world of contracting for the Metropolitan Transportation Authority, the agency that oversees the subway and is controlled by Mr. Cuomo.

After securing a $23 million contract to clear clogged subway drains in October 2017, William Haugland and his son Billy, the company’s leaders, sought to raise as much as a quarter of a million dollars for Mr. Cuomo’s re-election campaign at the Citi Field event, according to a person with direct knowledge of the event.

Neither the family nor their companies had ever donated to Mr. Cuomo before.

Haugland Group was suddenly one of Mr. Cuomo’s largest contributors as he ran for a third term . But it was hardly alone. Mr. Cuomo has raised millions of dollars from companies that work for the transit agency, benefiting from the fact that New York does not limit donations from contractors that do business with state entities.