AT&T Inc. boss Randall Stephenson told employees Friday it was a mistake to hire Trump attorney Michael Cohen and ousted the telecom’s giant’s top Washington executive after his office paid Mr. Cohen $600,000 last year.

“Our reputation has been damaged,” Mr. Stephenson wrote in a memo to staff. “There is no other way to say it—AT&T hiring Michael Cohen as a political consultant was a big mistake.”

Mr....