Can penicilln vk get rid of a reoccurring BV infection?

Unlikely. If you are having recurring BV you should check to see if treating the skin of your sexual partner's penis (if male) is colonized with these organisms. You need to discuss this with the doctor diagnosing and treating you. May need also to change the vaginal ph. Penicillin alone has a potential to work but is not the therapy of choice.