A weak eye pattern often is due to a weak laser, there are a few points to check before replacing the pickup or cranking the laser power up to get the last hours out of it.
As usual make sure power is clean first.
Play jumping at a fairly long regular interval as in your case could be due to dirt on the sled motor reduction gears.
If there was plenty of dirt on the lens make sure you cleaned it well enough, it would also be worth blowing under the lens (holding it to not tear it off it's hinges) with a can of dry air. Fine dust will also deposit on the prism under the lens.
It may be worth checking the power decoupling and solders on the focus/tracking coil drivers.
When I repair a (vintage) CD player I'll usually also check the spindle and sled Mabuchi motors using an oscilloscope with "component tester" function and 10X horizontal magnitude. Spinning the motor by hand will show the electric state of the motor, dirty brushes are usually fixed with contact spray.
I'll also try to check that the disc support has not moved down on the spindle motor axle. Sometimes this will stop the player from playing at all, other times it will play some disks and skip often, the focus servo will have a hard time. Only fiddle with this if you're really sure it has moved.
After that I'll try to get a disc to play, scope the eye pattern (If referring to an amplitude on the service manual make sure your scope/probe BW is sufficient) adjust the focus for highest amplitude, adjust the tracking for a compromise of highest amplitude without it being too noisy (acoustic noise).
Check with a CD-R.
Only after all this I'll consider playing with the laser power.