Michael Cohen has disavowed responsibility for some of the crimes to which he has pleaded guilty, privately contending in a recent recorded phone call that he hadn’t evaded taxes and that a criminal charge related to his home-equity line of credit was “a lie.”

As he prepares to begin a three-year prison term on May 6, Mr. Cohen, President Trump’s former lawyer, expressed dismay during the conversation that after testifying for more than 100 hours to federal and congressional investigators about his work for Mr. Trump—including...