I recently got a Fisher Price #816 battery-powered phonograph from 1983. It played fine for a week and then the speed started to intermittently increase to “chipmunk” speed without warning. Moving the volume control to certain positions also effected the speed. No fun!
I took it apart and found that the player has an electronic speed control, with trim-pots to fine-tune the 33 1/3 and 45 rpm speeds (VR1=680Ω, VR2=5kΩ). When I poked at the trim-pots the speed went crazy. They were probably oxidized. I marked their positions and swept them back and forth a few times to clean their tracks. That totally fixed the speed problem so I returned them to their marked positions. (Technically you should use a strobe disc to re-calibrate the speeds but it sounds close enough to me!)
The other battery-powered Fisher Price record players have the same internals as this one, so if your Fisher Price #820 or Fisher Price #3814 have speed problems then you might have dirty trim-pots. If you have the AC-powered Fisher Price #825 then this won’t help. (Those players have a simple tire-driven mechanism with different sized idlers for each speed. There is no electronic speed control.)
Also, I was surprised to find that my player had an internal switch that was activated when the tone-arm moved to the lock-groove at the end of a side. Strangely, this switch had no effect on the motor or amplifier (Instead it spins endlessly, depleting the battery). I’m not sure what this switch was intended to do, so I jumpered its pads on the PCB and rewired it as a power cutoff. The picture below shows the wiring before my modification.
This Old Toy has info about Fisher Price phonographs
This thread on Audio Karma has more photos of the interior of a Fisher Price #816
This thread on Antique Radio Forums has photos of the interior of a Fisher Price #820
The Portable Record Player Breakdown from Flea Market Funk summarizes battery powered phonographs