The private meeting last year between Gov. Andrew M. Cuomo and the Democrats in the New York State Senate was going as many did: Mr. Cuomo was pontificating and the Democrats were listening, neither side particularly pleased.

The Democrats were asking for the governor’s help in winning elections and breaking up a group of eight Senate colleagues who had crossed party lines to help keep Republicans in control of the chamber. But Mr. Cuomo suggested that the Senate Democrats’ problem was that they only understood New York City voters.

Senator Andrea Stewart-Cousins, the Democrats’ leader, interjected.

“You look at me, Mr. Governor, but you don’t see me,” she said. “You see my black skin and a woman, but you don’t realize I am a suburban legislator.”