The bottom readable surface is polycarbonate, aka "bulletproof glass." The data is on the other side, under the printing and is protected by the thick plastic. So you can use anything to clean the readable surface that won't melt or scratch polycarbonate. Eyeglasses are polycarbonate, so anything designed to clean glasses will work safely.
If the marks are caused by volatile outgassing from soft plastic sleeve then it might have etched the CD and is not washable. More likely, it just left residue, or glued itself to the CD and left some sleeve material behind.
I would start with dish soap and sponge. Milk is known as a very gentle cleaning solvent. Everyone knows that spit and fingernail will safely clean anything that's cleanable except germs. I would not use saliva after drinking milk, however.

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