WASHINGTON—President Donald Trump is relying on longtime outside advisers rather than White House staff as he deals with several major issues simultaneously, from a North Korean summit to the Russia investigation. But the perils of that approach were on display this week following a television interview he orchestrated with his new counsel, Rudy Giuliani.

Mr. Giuliani on Friday clarified remarks he had made two days earlier on Fox News about when Mr. Trump became aware of a 2016 payoff that his lawyer Michael Cohen made to...