Problem with weeds and black plastic

I am not an expert in gardening and I need your help. I have a corner area in my front lawn that is full of weeds, they are now attached to the black landscape fabric sheet underneath and when I try to pull the weeds I tear the fabric, I feel like I’m not getting anywhere. Is it better to lift up the black fabric completely and replace it with a new one or get rid of it? Is it better to cover with soil or chip woods? The area where I live is on a hill and it’s exposed to wind and sun .The ground is hard with a lot of stones, very hard to dig.