“Don’t you think that there’s something about public office that means that you have to have the trust of people, that we can’t have our politicians going around apologizing all the time?” he asked, noting that city employees like teachers are barred from tweeting photos to their students. “How are we going to have the stature as a city to tell that teacher that they can’t work for the City of New York when our mayor has done the same thing?”

Now Mr. Weiner was in full attack mode, his voice loud and hoarse. “Listen, if you believe that my personal failings disqualify me, don’t vote for me,” he said. “But, my friend, I’m not sure exactly what it is that you’re asking for.” He asked Mr. Strauss which candidate he supported. The answer? Bill de Blasio.

“That’s a surprise,” said Mr. Weiner, rolling his eyes, before telling Mr. Strauss he’d be happy if Mr. de Blasio even considered his ideas.

But Mr. Strauss interjected, arguing that the club’s goal was to bring new people into the political system. “It is very difficult for us to do that when he have politicians out there who are behaving this way,” he said.

Mr. Weiner interrupted, shouting over Mr. Strauss. “I’m gonna tell you something! Here’s what I recommend you do, if that’s your question, that’s your concern, here’s what I recommend you do: I recommend you say to them, ‘You know what? We have a lot of voices in the Democratic Party,’” he said, arguing that all ideas should be considered, no matter where they come from. “Ideas matter!”