Is it grammatically correct to write "forgive my being late to this discussion" as an alternative to "sorry that I'm late to this discussion"?
Forgive my being late ...
is a typical example of a grammatical feature known as a gerund:
gerund |ˈjerənd| noun Grammar
a form that is derived from a verb but that functions as a noun, in English ending in -ing, e.g., asking in do you mind my asking you?. [NOAD]
So, yes, that is a perfectly grammatical expression.
I'd change to to for, but otherwise it seems quite correct to me.
Forgive my being late for this discussion
Any of the following should be acceptable:
- Forgive my being late
- Forgive me for being late
- Forgive my late arrival
Probably most speakers wouldn't find your first version is ungrammatical as such, but it's certainly not the most common way of expressing what you want to say. If have a look, for example, at ngram data for "forgive my being" vs "forgive me for being", you'll see that the latter version is several times more common.