As he looks to heighten his national profile, criticizing the Republican health care bill and President Trump’s immigration policy, Gov. Andrew M. Cuomo of New York has been forced to confront a political schism far closer to home.

For five years, a group of renegade Democrats has enabled Republicans to control the State Senate, even though they are in the minority. Mr. Cuomo, a Democrat, has at times benefited from that strange reality: Having a divided Legislature allowed him to position himself as a deal-making centrist.

But after the election of Donald J. Trump, pressure has mounted on Mr. Cuomo to reunite his party.

Reunification was the agenda of a strategy session last month in Mr. Cuomo’s Midtown Manhattan office, attended by nearly two dozen Democratic state senators. When the discussion turned to how to best win elections, Mr. Cuomo suggested to the assembled lawmakers — many of them from New York City — that the leader of eight breakaway Democrats, Senator Jeffrey D. Klein, had a better understanding of the suburbs than they had.

That was all Senator Andrea Stewart-Cousins, the Senate minority leader who represents the suburbs of Westchester County, needed to hear.