But Mr. Cohen’s assertions left many questions unanswered, including whether the payment was truly a personal gift by him or whether he was reimbursed by some other party, like Mr. Trump or an associate of Mr. Trump.
In a brief interview on Wednesday, Mr. Cohen declined to answer questions about whether Mr. Trump had reimbursed him, whether the two men had made any arrangement at the time of the payment, or whether he had made any payments to other women or accusers of the president.