most heartfelt apology 最真誠的歉意
We offered our most heartfelt apology for having offended her.
我們為曾經冒犯她，致以最真誠的歉意。

remorse for sth/for doing sth 懊悔；非常遺憾；自責
I felt guilty and full of remorse.
我感到內疚，并且非常懊悔。
He was filled with remorse for not believing her.
他因為沒有相信她而懊悔不已。

I didn't mean to do sth. 我不是故意做某事。
I didn't mean to hurt you.
我不是故意要傷害你。

on purpose 故意，有意地
I didn't do it on purpose.
我不是故意這么做的。

not my intention 不是我的本意
It was not my intention to insult you in the way I did.
這樣傷害了你，并不是我的本意。

That was wrong of me. 我錯了。
My bad. 我的錯。