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Apologetically
Expressing Sorrow
Use 'apologetically' when expressing sincere regret for a mistake or action that affected others. 
Non-verbal Cues
'Apologetically' often accompanies gestures that reinforce the sense of being sorry, like a frown or downcast eyes. 
Subtle Humility
The adverb can convey a humble attitude, showing one does not assume they're deserving of forgiveness. 