Gov. Andrew Cuomo fended off a left-flank challenge from Cynthia Nixon in a Democratic primary Thursday, but several insurgents running on a slate with Ms. Nixon scored victories—lightly bruising the governor’s centrist brand but bringing him closer to a third term and preserving a path for a presidential run.

With two-thirds of precincts reporting, according to the Associated Press, Mr. Cuomo won 66% of the vote compared with 34% for Ms. Nixon, an actor best-known for her role in “Sex and the City.” Lt. Gov. Kathy Hochul...