It was shortly after that article was published that reports surfaced of a federal investigation into Mr. Weiner’s conduct. The F.B.I. found on Mr. Weiner’s laptop a trove of emails that belonged to Mr. Weiner’s now-estranged wife, Huma Abedin, a top aide to Hillary Clinton.
That discovery led to the late October announcement by James B. Comey, then the F.B.I. director, that the bureau had opened a new investigationinto Mrs. Clinton’s handling of official email, an inquiry that ended two days before the election. Mrs. Clinton has blamed Mr. Comey in part for her loss to President Trump.
In their court filing, Mr. Weiner’s lawyers, Arlo Devlin-Brown and Erin Monju, included a copy of a letter they sent last month to prosecutors, asking for “any information relating to political motivations by the victim or her father to damage the political prospects of Secretary Hillary Clinton.”
They also sought any evidence of “disclosures made to the Trump campaign or its surrogates” before the publication of the DailyMail.com article.
It is not known if or how prosecutors responded to Mr. Weiner’s lawyers, but the defense memo cites statements by the girl to investigators who interviewed her.
The memo says that the girl told the investigators that she “hoped somehow to influence the U.S. presidential election, in addition to securing personal profit.”
She also said she began communicating with Mr. Weiner because she was “looking for material for a book — one she has now written and is shopping to publishers,” Mr. Weiner’s lawyers wrote.
The memo says that the teenager told investigators that to generate material for her book, she “needed Anthony to act out — behavior she tried to elicit during their first exchange.” She suggested that they do a video chat, “immediately after volunteering to ‘prove’ that ‘I have a vagina,’” the memo says.
They began chatting online, with the girl “selectively taking screenshots of her phone to document the exchanges before they disappeared,” the memo says.
The claim that the teenager wanted to affect the election is not entirely new. In May, The New Yorker reported that Sydney Leathers, an earlier sexting partner of Mr. Weiner’s, sent him a private Facebook message on July 30, 2016, saying she had been contacted by the 15-year-old girl who had described exchanging illicit messages with Mr. Weiner.
“She’s talking about potentially messing with Hillary’s campaign,” Ms. Leathers told Mr. Weiner.
According to the New Yorker article, Ms. Leathers put the teenager in touch with DailyMail.com, which published her story in late September.
Mr. Weiner is to be sentenced by Judge Denise Cote on Sept. 25. Mr. Devlin-Brown, his lawyer, declined to comment.
A lawyer for Ms. Abedin declined to comment on Mr. Weiner’s sentencing submission. On Wednesday, she and Mr. Weiner appeared before a Manhattan judge who was hearing their divorce case.
Mrs. Clinton’s lawyer also declined to comment. In her new book, “What Happened,” Mrs. Clinton described learning about Mr. Comey’s new inquiry, and how it stemmed from the unrelated investigation into Mr. Weiner.
“When we heard this, Huma looked stricken,” Mrs. Clinton writes. “Anthony had already caused so much heartache. And now this.” She recalls that Ms. Abedin burst into tears and said, “This man is going to be the death of me.”