With Medicaid costs soaring in New York, the Greater New York Hospital Association was pushing for the seemingly impossible: more state reimbursement money .

It was a big ask, and for years, it had gone nowhere. Medicaid spending already represented an enormous and ever-growing share of the state budget, and Gov. Andrew M. Cuomo had taken steps to keep the program in check.

Then things changed.

As Mr. Cuomo was locked in a bitterly fought Democratic primary last year, his campaign asked the association, one of Albany’s most influential and richest power centers, to make a major donation to the State Democratic Party, according to a person familiar with the discussions.

The hospital association wrote two checks for the state party, totaling more than $1 million, campaign finance reports show. It was twice as much as the association, which represents New York City’s biggest health care institutions , had given to any campaign in at least a decade.