If you have to cover it with something to hold it down, why not just mulch with wheat straw. Someday you might wish that fabric cover was never used.

Can make a hooded sprayer by cutting the bottom out of a milk jug and inserting spray wand through small hole in lid.

Pull broad leaf weeds and spray grass with clethodium.

Prowl or metolachor can be used to help prevent weeds from coming up.