CD player cuts out

(((Try not to use alcohol or Windex on any optics you may have in the house, they do attack glue or glass, at least in microscope objectives. Get a bottle of Zeiss lens cleaner off Amazon)))

I fix Discmans as a hobby, and in most cases this skipping, more pronounced at the end of a disk, means worn spindle motor. Since CD drives for CD50 were discontinued decades ago, you may try to reset CD platter onto a new Mabuchi motor.

Alternatively, if only some specific CDs skip, then you are in a good company of "post-VRDS" Wadia owners ;-) learn to live with this glitch... You may find this post (at square-2.com) interesting:
THE GOOD, THE BAD AND THE UGLY. CD PRESSING QUALITY


I give up with the link!!! here is the address:

http://square-2.com/the-good-the-bad-and-the-ugly-cd-pressing-quality/

A hair may be stuck to the laser sled. Have you opened up the top and taken a look? Another reason may be the laser sled rails are dry.
Lasers move around on thin straight round metal rails under the lens assy. they 'stick' a tiny bit and that messes up the sound. Often one can just open the cover, there will be white grease at the ends of the rails, use a toothpick and just move SOME arund on the rails. Do not use all of it. So next time you have some left. You can also buy the stuff. 
Molykote EM-30L or a white lithium grease. But the Molykote EM-30L is the right stuff.

+1 for checking sled gears and sleds, if you are lucky that's might be it!
Forgot to add: before opening anything check how it plays CDR and CDRW first, it may give you a better idea on what is going on.