Every record collector sufferes from a skipping record one day. Even after cleaning it with a professional cleaning machine, skips are still recognizable sometimes. Since I have tried to fix it with a wooden toothpick I have had succeeded to fix most of the skips from old unplayable records.
Here is what I've done:
Step 1:
Clean your record like you normally do. This can be done manually by using a cloth or a cleaning machine. Usually isopropanol mixed with distilled water (1:3) is a good way to get worst dirt off of your record.
Step 2:
Play your record until you notice the skip and stop it exactly where it skips.
Step 3:
Use a wooden toothpick and rub the head of it gently over in direction of the groove a couple of times.
Step 4:
Play the groove that skipped and look if you've cured the fault. If not try it again or take the record off of the turntable and repeat the procedure with a magnifying glass.
Step 5:
Clean your record again (step 1)
In most cases that helped me to repair the skip even though you can still hear some noise when done with deep scratches.
Try it and happy listening!