NEW YORK (CBSNewYork/AP) — Mayoral candidate and former U.S. Rep. Anthony Weiner responded apologetically Tuesday following revelations of another alleged cybersex relationship, but said he is not going anywhere when it comes to the mayoral race.

“I want to bring my vision to the City of New York,” Weiner, a Democrat, said at a news conference with his wife, Huma Abedin, by his side late Tuesday afternoon. “I hope they are still willing to give me a second chance.”

Anthony Weiner And His Wife Address The Latest Sexting Allegations: Watch The Press Conference Here

Weiner: 'I Am Very Sorry' For Latest Sexting Revelations

Weiner began the news conference by reading a statement he had released earlier.

“I have said that other texts and photos were likely to come out, and today they have. As I have said in the past, these things that I did were wrong and hurtful to my wife and caused us to go through challenges in our marriage that extended past my resignation from Congress. While some things that have been posted today are true and some are not, there is no question that what I did was wrong,” Weiner said in his statement. “This behavior is behind me. I’ve apologized to Huma and am grateful that she has worked through these issues with me and for her forgiveness. I want to again say that I am very sorry to anyone who was on the receiving end of these messages and the disruption that this has caused. As my wife and I have said, we are focused on moving forward together.”

Weiner would not go into details about the specifics of the claims.

“One of the things I’m not going to do is I’m not going to get into a back-and-forth with people about these things and whether they are true or not,” Weiner said.

But his tone was contrite.

“Let me reiterate how sorry I am to the people that got these messages, for any inconvenience or embarrassment they caused,” he said.

Abedin also spoke about her husband and his scandals.

“Really what I want to say is, I love him. I have forgiven him. I believe in him. And as we have said from the beginning, we are moving forward,” she said.

Abedin said it was “not an easy choice” to stay with her husband, but she made the decision to do so.

“I didn’t know how it would work out,” she said. “But I didn’t know that I wanted to give it a try.”

She conceded that her husband had made “some serious mistakes – both before he resigned from Congress and after.”

In speaking to reporters, Abedin was calm and smiled occasionally, but began her remarks by saying she was nervous because she had not appeared at such a news conference before.

Weiner said repairing his relationship with his wife – a former aide to Hillary Clinton – was more important than his choice to resign from Congress.

“I’m glad these things are behind us,” he said.

The latest sexting allegations hit the headlines when a woman claimed anonymously this week to the gossip site “The Dirty” that she exchanged lewd photos and had explicit phone conversations with Weiner. She said the messaging took place after the scandal that led him to resign from Congress in 2011.

The woman told the website she was 22 when Weiner began the online relationship. Site editor Nik Ritchie wrote that the woman claimed Weiner and the woman “had a relationship for six months, and she believed they were in love.”

The relationship reportedly continued into last summer, CBS 2 ‘s Marcia Kramer reported.

Baruch College School of Public Affairs professor Micheline Blum said the new allegations do not reflect well on Weiner.

“This is bad because of the timeline, because this happened after he resigned – and presumably when he was apologizing and in treatment, and it was supposedly over,” she said. “That can’t help, and it’s got to make New Yorkers feel that maybe there’s more out there.”

Weiner allegedly used the pseudonym “Carlos Danger” in the conversations, the website reported.

In one of the cleaner sexts, Weiner reportedly tells the woman: “Sadly my pics are out there to look at. Have you ever?”

The woman replies: “Yes. And I must say I was quite impressed.”

“You are a walking fantasy,” Weiner says in another reported text.

“I don’t want to be just a fantasy,” the woman replies. “I want to take care of your every need.”

Many of the other texts went into graphic details about imagined sex acts between Weiner and the woman – using explicit terms for genitals and sex acts.