Student or Learner
I saw a vedeo clip in which Paul McCartney speaks to his fans in Japan.
He said, "Yeah, to all you people who've been looking forward to me coming back to Japan...." (He has been to Japan several times.)
In his comment, Paul said, "...who've been looking forward to me coming back to Japan", but grammatically, I think it is also fine to say "who've been looking forward to my coming back to Japan". If it is fine, does it sound diffrently, like it's politer or old-fashined?
In my experience, fewer and fewer people are using the possessive before gerunds these days. And almost no one objects to using the object pronoun by others, even if they use the possessive themselves.
Using "my" is "correct" under the older rules of grammar. But educated people will use "me" and not worry about it.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.