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I don't hope to gain your forgiveness; I just hope you will feel better after my apology.
I'm not begging your forgiveness ...

Hi,
Do both of the above sound good to you, especially "gain your forgiveness?" Thanks.
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Gain forgiveness is OK.

I don't think the meanings are the same. Not to hope and to not be doing something are different.
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Hi Angliholic

I've heard people say 'to beg/ask for your forgiveness' and 'to get/have your forgiveness'.

PBF

[edit] Deleted 'I don't think people say 'to gain sbd's forgiveness' because apparently people do say 'to gain forgiveness'. My apologies.
 Mister Micawber's reply was promoted to an answer.