Hunter Biden settled a child-support dispute with Lunden Roberts, the mother of his fourth child, on Thursday, ending a yearslong saga that had become tinged with partisan politics.

According to court documents, Mr. Biden, 53, agreed to pay a monthly sum, which was not disclosed, to Ms. Roberts, as well as turn over several of his paintings, the net proceeds of which would go to his daughter. Mr. Biden, who is in recovery from a crack cocaine addiction, started a second career as a painter whose works have been listed for $500,000 each.

He also promised to discuss planning for a college education fund with his child’s mother. Ms. Roberts, 32, who had requested to change the girl’s last name to Biden, dropped the request as part of the agreement.

The agreement brings to a close a case that shined a spotlight on Mr. Biden’s personal behavior and finances and was used by critics to attack his father, President Biden.

Hunter Biden had been paying $20,000 a month in child support for several years, for a total of $750,000, according to his attorneys. He had argued that he was not financially able to support the original child-support order. The new amount is lower than had been originally ordered, according to a person familiar with the case.

But Ms. Roberts and her attorneys argued that the girl, 4, was entitled to the same benefits that her four half-siblings receive.

Ms. Roberts and Mr. Biden met in Washington. In mid-2018, Ms. Roberts was working as his personal assistant, according to a person familiar with the case. Their daughter was born later that year, but by then, Mr. Biden had stopped responding to Ms. Roberts’s messages, including one informing him of the child’s birth date, according to a person involved in the case, who asked not to be named because of the sensitivity of the matter.

Ms. Roberts filed a lawsuit in May 2019, and DNA testing that year established that Mr. Biden was the father of the child. In a motion for custody filing in December 2019, Ms. Roberts said that he had never met their child and “could not identify the child out of a photo lineup.”

A bench trial had been scheduled for mid-July in a court in Arkansas, where Ms. Roberts lives, and her attorneys had enlisted Garrett Ziegler, a conservative activist who had compiled the contents of Hunter Biden’s laptop into an online database, as an expert witness with knowledge of Mr. Biden’s finances.

Two weeks ago, when Mr. Biden arrived in Little Rock to deliver a deposition, Ms. Roberts attended. According to a person familiar with the proceedings, Mr. Biden’s attorneys argued that Ms. Roberts had saved only a small amount of the $750,000 Mr. Biden had paid her for their child.

They also questioned why, considering the conservative figures involved in the case — including Mr. Lancaster, who had been an attorney for the Trump campaign during an electoral vote recount in Wisconsin after the 2020 election — she would want the Biden last name.

Ms. Roberts did not immediately respond to a request for comment, but Mr. Lancaster said in a text message that Mr. Biden attorneys’ claims were “not true.”

“The case settled because, one hour into Hunter’s deposition, he asked to speak to Lunden privately,” Mr. Lancaster continued. “They spoke for approximately 45 minutes to an hour. They emerged with a settlement.”

He added that, going forward, Mr. Biden’s child would be selecting art from her father, and that family was “very important” to his client.

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