"After annoying them" or "After having annoyed them" but NOT "After having annoying them" I can't think of a case where it would be appropriate to use "having" + V-ing.

thank you so much :)

After 'have' you either have to have a noun (to indicate ownership) or a past participle (to indicate present perfect). You can never say 'have verb+ing' or 'having verb+ing'. If you saw that form, maybe the second word was an adjective ending with 'ing', which should have been followed by a noun.

I mean "difference" not diffenrence, sorry about that :)