Weiner asked the judge for probation and community service.
Attorneys for the disgraced former city Congressman say Weiner should be spared from prison because of a "deep sickness".
"'Whatever else Anthony has done, he loves" their son, Abedin wrote of Weiner, who is a Jewish and who in May admitted to sending obscene material to a 15-year-old girl from his cellphone. "As she later stated to Government investigators, she also hoped to somehow influence the U.S. presidential election, in addition to securing personal profit".
On Oct. 28, former FBI Director James Comey sent a letter informing Congress that agents were reviewing the emails "to determine whether they contain classified information" relevant to a separate investigation of Clinton's use of a private email server while Secretary of State.
Weiner can receive up to 10 years in prison. There is no mandatory minimum prison sentence for the charge but prosecutors have requested sentence between 21 months and 27 months.
The FBI seized Weiner's laptop and found a large trove of emails belonging to his now-estranged wife, Huma Abedin, a top aide to Clinton.
The investigation into Weiner's exchanges with the teenage girl roiled the 2016 USA presidential campaign in its final days, when authorities found emails on Weiner's laptop from Abedin.More news: London police condemn Trump's 'loser terrorist' tweets following tube 'bucket bombing'
Weiner notes in his letter that he wants to be the one to break the news.
The Daily Mail story (for which the girl ultimately received $US30,000, the lawyers wrote) "jolted Anthony in a way other scandals had not, and led him for the first time to grapple with the depths of his sickness and to find the strength to treat it". "My continued acting out over years crushed the aspirations of my wife and ruined my marriage".
"This is not a letter I ever imagined I would write, but, with Anthony, I have repeatedly found myself in circumstances I never imagined", Abedin wrote in her letter, the public version of which was heavily redacted. "I am devastated by Anthony's actions, and I understand he must face their consequences", Abedin wrote.
When his laptop was seized by the Federal Bureau of Investigation, it was discovered Ms Abedin had used it to send thousands of emails to Mrs Clinton's private account.
Mr. Weiner is to be sentenced by Judge Denise Cote on September 25.
He said "the one flawless thing in my life - my son - will forever have to answer questions about the public and private failings of his father". Mrs. Clinton has blamed Mr. Comey in part for her loss to President Trump.
"Seeing that Weiner is both a repeat offender - his sexting addiction cost him his job in Congress as well as a shot at becoming mayor of NY - and associated with one of the most important people in Clinton's inner circle, it is conceivable that this was a set-up from the beginning, with the objective of embarrassing the Clinton campaign", according to the WhoWhatWhy report. Weiner resigned from Congress in 2011, after the first of several sexting scandals came to light - the first with an adult woman through his Twitter account.