I have successfully done it a couple of times. I don't claim any special skills just the right equipment, patience and a steady hand.
I used:
1. Jewelers illuminated magnifying glasses, to keep both hands free, I used 8.3X magnification.
2. A belt driven turntable that freely moves both directions
3. A soft-wood toothpick. The harder the tooth pick the likelier you'll just end up further damaging the record.
The hardest part is identifying the groove where the needle was sticking. On both records that I fixed, I found a particle stuck the groove which was visible using 8.3X magnifying glasses. I gently rubbed the business end of the wooden toothpick on the area and was able to remove the particle. Did a good cleaning afterword using my VPI record cleaning machine, it did leave a smudge mark on the vinyl but no audible tick.
This will only work if there's a particle stuck in the groove, obviously. If the plastic is damaged there's likely nothing that can be done to fix it.
