Among the allegations are that one female worker was demoted after refusing to "become intimate" with a foreign government official.
The woman's manager reportedly made "sexist and suggestive comments" about her to the official.
Bono told the Mail on Sunday newspaper: "We are all deeply sorry. I hate bullying, can't stand it."
The newspaper said the charity publicly admitted the issues hours after it had sent a long list of allegations at the end of an investigation, in order to allow One to comment.
