A key state lawmaker criticized New York Gov. Andrew Cuomo’s plan to authorize mobile sports betting through a competitive bidding process, highlighting the obstacles ahead for legalizing online gambling in the state.

State Sen. Joe Addabbo, a Queens Democrat who chairs his chamber’s Racing, Gaming and Wagering Committee, said in an interview that he was glad the Democratic governor had shifted away from his opposition to online betting but believed the state should enact a more expansive system.

“I am not a believer that one sportsbook provider, or operator, can handle the volume in New York,” Mr. Addabbo said. “I don’t think it’s good for the consumer—I think you’re pushing the consumer to New Jersey or to an illegal site.”

Sports wagering in New York is currently allowed only in physical portions of its four existing commercial casinos and other facilities operated by Indian tribes.