1. in a manner expressing regret, sorrow, or acknowledgment of a mistake or offense
2. in a manner that shows embarrassment or self-consciousness about something

Examples

• She apologetically explained that she had forgotten the meeting.
• He smiled apologetically when he realized he had stepped on her foot.
• The waiter apologetically informed us that the kitchen was out of the special.
• She spoke apologetically about her late arrival to the party.
• He apologetically offered to pay for the broken vase.
• The manager apologetically addressed the customer's complaint.