No victory: How to apologize
Manda'ade don't offer empty apologies, but they do own up to their mistakes. Note that there is very little information about what Mandos say to each other as a way of apologizing or asking forgiveness. One feature of their culture that we do know is that they are very direct, so it is reasonable to assume that they only apologize if they know that they have wronged the other person.
For a good overview of when and how to apologize, plus some other alternatives to saying "I'm sorry," check out this article from Mulligan Advisors, a conflict management company.
See also Mercy and Kindness, as well as this page for different ways to express gratitude.
Here are some ways to apologize beyond simply saying "I'm sorry," which can show that you are paying attention to the situation.
In English
Thanks for the insight.
Thanks for the clarification.
That's an interesting way of looking at it.
Thanks for the new perspective.
I didn't go into enough detail, apparently. Let me clarify ...
I understand, and I'll keep it in mind in the future.
I see I made a mistake. How can I fix it?
(Some people don't bother with the last bit if they know what went wrong and how it should have gone. Instead, they fix the mistake, such as rewording a post if they can't get away with deleting it.)
In Mando'a
Spoiler pipes added for convenience in using in Discord servers.
Suvari. ||Understood/I understand.||
(For when someone corrects misinformation or a misunderstanding that you had about something.)
Ni suvari bal partayli ibac. ||I understand and remember that.||
(A longer way to way to accept correction.)
Ni gedeteya par din'kartay. ||I am thankful for the information.||
(A longer way to way to accept correction.)
Ni gedeteya par evaar'la mirde. ||I am thankful for the new thoughts.||
(A longer way to way to accept correction.)
Ni ami. ||I change.||
(For when you want to show you are committed to doing better. Use a fan word for "change.")
Gar serim. ||You're right.||
(For when someone provides a clarification for or expansion on something that you said that was otherwise correct.)
N'eparavu takisit. ||I eat my insult.||
(Spoken to someone after they make you aware that your actions hurt them, but not so bad that you need to beg for forgiveness. Because the subject pronoun "ni" was contracted with the verb, this must be a frequently used phrase in-universe.)
Ni aala trikar'la par ner nare. ||I feel sad for my actions.||
(An alternate to "I eat my insult." Or use narise as the plural for the word for actions.)
Ni nari dush. Tion nare jate'shya? ||I act badly. What actions are better?||
(An alternate to "I eat my insult.")
Ni nu dajuna vaabir ibac. ||I didn't plan to do that.||
(A way to express regret over an action that didn't work out.)
Ni nu vaabi mirgaanla. ||I didn't do (that) deliberately.||
(A way to express regret over an action that didn't work out.)
Ni nu nari mirgaanla. ||I didn't act deliberately.||
(A way to express regret over an action that didn't work out.)
Ni ceta. ||I throw myself at your mercy.||
(A rare form of apology reserved only for when you've done something that is beyond repair.)
Instead of "I'm sorry to hear that," consider "I feel sad for you" or "I hope things turnaround soon" or something similar. In Mando'a, use trikar (sadness) as a statement, ni aala trikar par gar (I feel sadness for you), vercopa nakar'tuur jate'shya (I hope tomorrow is better), or something similar.
Apologizing for a small interruption, bumping into someone, or similar small errors: In English, use "excuse me" or "pardon me." In Mando'a, use gedet'ye (please) or gedet'ye digu (please forget).