Michael D. Cohen, President Trump’s former lawyer, released a secret recording on Tuesday of a conversation between Mr. Cohen and Mr. Trump in which Mr. Trump appears to have knowledge of payments to a former Playboy model who said she had an affair with him.

The recording provides details on payments to the former model, Karen McDougal, but it does not definitively answer the question of whether Mr. Trump directed Mr. Cohen to make the payments in cash or by check.

The recording’s release contradicts repeated statements from Mr. Trump and his aides, and it shows how those in Mr. Trump’s orbit have used falsehoods to try to protect themselves from unflattering coverage and tough questions.