OK. I think I can help. There are two very simple solutions to a skipping CD player. You have tried one already, that is, clean the laser-reading eye. Lets assume you did that correctly and that is not the problem.
So, to continue, the machine must be opened. Equipment required: Jewellers screwdrivers, small number one screwdrivers, cottonswabs or Q-tips, Petroleum jelly, 100% isopropyl alchohol is ok but if you can use Denatured alchohol that is the best.
Using a set of fine jewelers screwdrivers or just a small #1 Regular or Philips screwdriver to remove the top cover of the CD player. Don't force anything and place the screws in a cup so they are not lost or misplaced. I work in a plastic cafeteria tray to contain any dropped screws
The problem here usually occurs in trusty old CD players that have always worked fine and the problem (skipping) starts out small and gradually progresses to always skipping. The problem and solution are mechanical in nature.
First locate the laser eye and check it physically for any optical obstructions (dust, dust-bunnies, lint, feathers, hair, lumber, etc.)
When you are sure the laser-eye is clean find two parallel small (usually 1/8" or about 3mm wide by about 3 to 6 inches long ) chrome metal bars that the laser eye rides back and forth on. The eye assembly has a small contact-lens looking clear plastic lens mounted on a plastic carriage that moves from the center of the disc toward the outside as it tracks the disc. It will have a small ribbon wire harness that can expand and retract with the travel of the laser-eye carriage. Now look at the two-parallel shiny metal bars the carriage should slide on. The lubricant on these bars lets the eye assembly move smoothly without vibration (or skips) over the length of the two steel bars and your CD plays perfectly when they are clean and lubricated. This skipping problem can occur if something sticky is spilled into the unit.
Soda or beer dries out and if it gets on these steel bars the laser-eye assembly can vibrate and chatter as it tries to track over the CD's info causing skips.
So the solution involves carefully testing the unit with the lid off to see what is going on inside when the unit plays (or attempts to play) a CD. If the obstruction to the path of the eye assembly is severe, the tiny drive gears can be damaged, broken, or stripped and the eye will not move smoothly without new parts (Factory Authorized Service Personnel are recommended in that case)
Use the Denatured Alchohol and swabs to completely clean and dry the steel track rods that the laser - eye slide on, take care to clean any built up old grease (manufacturer lubricant) that may build up on the holes on the sides of the Laser-eye assembly where the steel track rods go through it. After cleaning the rods use a new swab to apply a TINY amount of petrolatum evenly to each rod. I cannot stress enough how important it is to use as little lubricant as possible ONLY on the rods and NOWHERE ELSE! It is very easy to get the petrolatum somewhere you don't want it. Be Careful! One mistake here could cost you the entire unit! after application, physically slide the eye assembly over the bars to smooth out the petrolatum lubricant evenly. (in the event the eye-assembly doesn't manually slide, don't force anything and go to the next step). make sure you did not get any petrolatum where you did not want it, if you did clean it with cotton and good old, denatured alcohol and put a CD in and before playing it, advance the tracks from track 1 to the highest numbered track and back several times to smooth the lube more. Play the CD and replace the lid and screws in the order you removed them.