Actually, it depends... (Please for give the paraphrasing - it's for brevity and simplicity.)
quote:"...of my correcting his grammar." Gerund.
I can relate. My boyfriend is very, very sick of me correcting his grammar.
"My boyfriend is sick of me correcting him."
In this sentence, you could argue that the phrase "me correcting him" is a noun, as in, "Here is a photo of me correcting him." "Correcting" in this context is not a gerund (a verb made into a noun by adding -ing, for those who don't know) but a participle verb.
"My boyfriend is sick of my correcting him."
On the other hand, the "my" is an adjective describing "correcting". Since, as you pointed out, "correcting" in this context is a gerund, "correcting him" is ungrammatical, since they are both nouns (well, a noun and a pronoun).
By my linguistic reckoning, if you want to put the "my" in there, the sentence should be this:
"My boyfriend is sick of my correction of him."
...or, in the phraseology of the original quote:
"My boyfriend is very, very sick of my correction of his grammar."
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