Some years ago, I saw something called "Disk Doctor", "Game Doctor", or a similar variation, which was a gadget to spin your CDs with some kind of solution which would polish them to get rid of small scratches. Some used CD stores have a "professional" version of the same idea, a machine that resurfaces the disc. But it can't solve all problems -- deep scratches are sometimes unfixable, and require new copies.
Because a CD is basically a continuous stream of data, in a spiral from the center outward, a lateral scratch (going sort "with the grain" of the spiral) is much worse than a transverse scratch (radiating outward from center to edge) because it wipes out a whole bunch of data in a row, while the transverse scratch sometimes only wipes out small bits, which can be recovered by the built-in error correction of the CD standard.
So, when wiping CDs to clean them, always wipe radially, in case a small piece of debris is in your cleaning chamois and gets rubbed against the disc surface.