Weiner's video, posted shortly after midnight, asked New Yorkers "to give him a second chance to work for you." While it ended with a political pitch, much of the video showed Weiner with his wife, Huma Abedin, who worked as an aide for Secretary of State Hillary Clinton, and their son and reminded voters of his roots in Brooklyn as the son of a teacher and a lawyer who got his degree on the GI Bill.
In 2011, Weiner, who represented a New York City district as a Democrat, resigned from Congress in disgrace. He had been caught sending explicit photos to women who followed his Twitter account.
"I made some big mistakes and I know I let a lot of people down but I have also learned some tough lessons," he said on YouTube.
Abedin also pushed her husband's candidacy.
"We love this city and no one will work harder to make it better than Anthony," she said.
At least three other candidates are seeking the Democratic nomination, the New York Daily News said. Council Speaker Christine Quinn, the front-runner, has been slipping, the News said, and Public Advocate Bill de Blasio and Bill Thompson, the city's former controller, are also in the race.
A poll last month put Weiner in second place.