most heartfelt apology 最真诚的歉意
We offered our most heartfelt apology for having offended her.
我们为曾经冒犯她，致以最真诚的歉意。

remorse for sth/for doing sth 懊悔；非常遗憾；自责
I felt guilty and full of remorse.
我感到内疚，并且非常懊悔。
He was filled with remorse for not believing her.
他因为没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.
我不是故意要伤害你。

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

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

That was wrong of me. 我错了。
My bad. 我的错。