If the scratch is wide enough it may prevent the laser from reading the recorded data. The inventors of the CD formats knew there would be errors from time to time. So they embedded error correcting information into the digital tracks. Cool idea, except when a tangential scratch destroys a significant portion of the disk. There are simply no error codes left to correct the data.
Radial scratches are different.
Because they run perpendicular to the circular tracks, only a small chunk of information may be lost. Error correction can easily compensate for tiny amounts of lost data. Cleaning solutions should be chosen with care or not used at all. CDs are made from a tough polycarbonate plastic and should live through common stain or gum removing solvents. Experiment on an unimportant CD first.