Cork Backing? The Roof Liner is a fibrous material flexible shaped board, haven't seen any cork in it. You shouldn't see any cork.
There is the board, that has "hood liner" cloth glued to it. The "Hood Liner" cloth is the flexible cloth with a thin layer of cushioning foam attached to the back.
Usually the thin layer of cushioning foam breaks down and lets the cloth portion sag. You can go for the cheap/ugly/quick fix and just poke some tiny holes and spray some adhesive in the sagging portions and push it back up. Since the cushioning foam is all gone, your bound to see all the cracks/imperfections, plus put in wrinkles, etc. It will be ugly, but get the cloth up and out of the way.
You can take the whole headliner down, board and all, strip it and re-glue new fabric on it. That takes a whole afternoon, but looks much better.
You can find hood liner fabric at any big fabric store/outlet. I got enough to do my roof for $30 at Jo-Ann Fabrics.
You could use cheaper clothe if you choose, but it won't have the layer of cushioning foam behind it. That cushioning foam hides all the imperfections on the backing board, that will show up with regular cloth. Plus the cushioning foam gives the headliner that nice feel to it, regular clothe might not only look bad, it also may feel a bit rough and stiff to the touch.