Use 'apologetically' when expressing sincere regret for a mistake or action that affected others. He apologetically admitted that he had lost the keys.
Pro Tip 2/3
Non-verbal Cues
'Apologetically' often accompanies gestures that reinforce the sense of being sorry, like a frown or downcast eyes. Dropping her gaze, she spoke to them apologetically.
Pro Tip 3/3
Subtle Humility
The adverb can convey a humble attitude, showing one does not assume they're deserving of forgiveness. He apologetically asked for a second chance at the interview.