IPA: /ˌæpəˈlɒdʒɪkli/
KK: /əˌpɑlədʒˈɛtɪkli/
In a way that shows you are sorry for something you have done or said.
She spoke apologetically after realizing her mistake.
Apologetically → It is formed from "apologetic" (from Greek "apologia", meaning a speech in defense) and the suffix "-ally" (meaning in a manner of). The word "apologetically" means in a manner that expresses a defense or justification for something.
Think of making a 'defense' ('apologia') for your actions and doing it 'in a manner' ('-ally') that shows you are sorry or regretful.