1. I hope you don't mind that I'm X-ing you with Y.
2. I hope you don't mind the fact that I'm X-ing you with Y.
3. I hope you don't mind my X-ing you with Y.
4. I hope you don't mind me X-ing you with Y.
All the above are possible. #1 is quite formal; #2 is quite formal and wordy; #3 is moderately formal; #4 is neutral.
#4 is probably the commonest form, in ordinary English.
However, some people would prefer #3 to #4, on the grounds that "X-ing you with Y" can't be said to qualify "me", since "I hope you don't mind me", in isolation, would have a different meaning from its meaning in the sentence.
(I would disagree; "X-ing you with Y" looks like an idiomatic object complement, to me.)
All the best,
Student or Learner