Most likely the record needs tightened if it wobbles. The larger plastic nut is more for raising or lowering it so it sits flush with the playfield.
There is a small 1/4” screw head on the bottom under the playfield. And another small Phillips under the record label. It’s possible to press super hard on top of the record to hold that Phillips and then use 1/4 bit driver for the screw on the bottom to tighten it so you don’t have to puncture the label but it’s difficult and may work loose again. I sort of press that with my thumb while wedging my fingers around the post the ball hits to get my leverage or hold it steadier.
I did it that way for a long time but I finally just punctured my label recently to get it tighter, for longer.
I pull the playfield to service position, then get under with the bit driver and get it on that nut. You can get in there from the side with the bit driver in one hand while you hold the record from the top.