Yes, you can say "my / your/his/her/ our/ their English". You can say I have improved my English" The last sentence sounds strange to me. John

Yes, you would definitely say "my/your English." "I have improved my English" is perfectly correct. If you simply wrote, "I want to improve English", that means you plan to revise the whole language for other people. It's not about your own skills in the language. The last sentence could be written as "I have improved myself through English", but this means "I have become a better person in general, and English helped me to do this." Maybe you were thinking, "I have improved at English."