President Trump’s former lawyer told Congress on Wednesday that Mr. Trump remained deeply involved in the arrangement to pay hush money to a pornographic film actress, even while in the White House, telling him just a year ago to lie to a journalist about the president’s involvement.

Testifying before the House Oversight Committee, the lawyer, Michael D. Cohen, also for the first time publicly implicated the president’s eldest son, Donald Jr., and the chief financial officer of his company, Allen Weisselberg, in the arrangement involving the actress, Stormy Daniels. She had been paid $130,000 before the 2016 election to keep quiet about an alleged affair with Mr. Trump, and Mr. Cohen was reimbursed in 2017 for fronting that money.

“He knew about everything,” Mr. Cohen said of Mr. Trump. “Everything had to go through Mr. Trump and had to be approved by Mr. Trump.”

While federal prosecutors in New York had laid out a general narrative around the payments, including Mr. Trump’s involvement, Mr. Cohen explicitly placed Mr. Trump at the center of the events, producing checks he described as evidence of the role the president and his company played.