Hi kaori. Yep you’re right you don’t have to use my in the sentence but it does make the sentence flow better. You could say something like “I’ve been trying to get better in English these last few months with language partners” and I would understand you perfectly. I would try to stay in the habit of using descriptive and personal words when talking about yourself though (I, my, mine, etc) so you can understand and learn when you do and don’t need them. Hope this helps

I have been trying to get better in English these few days with language partner