“The Westchester seat is the one where the most money and attention are being spent,” said Geoff Berman, executive director of the state’s Democratic Party. “We’ve been doing all sorts of phone banks and organizing canvassing trips from around the state to help Shelley.”

The big challenge with special elections, of course, is voter turnout. Unlike the November elections, special elections can escape the notice of all but the most avid followers of politics. In the past year, more Democrats have become involved in local politics than any time in recent memory. “The Democratic base is a lot more energized since the election of President Trump,” said George Picoulas, a lecturer in political science at Pace University.

In the Assembly, where Democrats have a nearly three-to-one advantage over Republicans, the nine elections on Tuesday could afford Republicans the opportunity to make inroads and the Democrats to solidify their grip. Five of the nine seats were held by Republicans, including all three on Long Island and two upstate.

Two fresh vacancies in the Assembly, resulting from two certainties of life in Albany — death and indictment — will be decided in the November election.

The three Assembly seats in New York City are expected to be filled by Democrats. In the 39th District in Queens, which includes Jackson Heights, the only major-party candidate to appear on the ballot is Aridia Espinal, a Democrat and former aide to Francisco Moya, who left the Assembly for a spot on the City Council.

In the 80th District in the Bronx, which includes Pelham Gardens, Mark Gjonaj also quit the Assembly for the City Council. The Republican candidate, Gene DeFrancis, is a United States Navy veteran who founded a merchants association. He will face Mr. Gjonaj’s former chief of staff, Nathalia Fernandez, who also worked as a Bronx representative for Mr. Cuomo.

The recent move of Brian Kavanagh, a Democrat, from the Assembly to the Senate opened up the seat in the 74th District on the Lower East Side of Manhattan, where Harvey Epstein, a lawyer and former community board chairman, has the Democratic nod. He will oppose Bryan Cooper, an event planner and perennial Republican candidate.