David Letterman apologized on the air Monday to his wife, Regina, for his revelation last week of his sexual relationships with staff members, saying she had been “horribly hurt” and adding, “I’ve got my work cut out for me” to mend the relationship.

Mr. Letterman made the remarks at the start of the taping of his Monday night show on CBS. They were his first public comments since last Thursday when he revealed details of an apparent extortion attempt tied to sexual relationships with his staff members.

The comedian also apologized to his staff, saying, “I’m terribly sorry that I put the staff in that position.” He labeled his mistake as inadvertent because “I just wasn’t thinking ahead.” He thanked the staff for being supportive and for “putting up with something stupid I’ve gotten myself involved in.”

In his monologue, a rueful Mr. Letterman told several jokes related to the incident and the critical reaction to it.