My guess as i have come across this so many times... Broken Shift Drum Stopper... Its the little spring loaded arm with a wheel on the end that clicks into the notches on the star looking end of the shift drum. Usually symptoms are it will pop out of gear or it will only go into first and you can't advance because the stopper isn't catching the shift drum for the next push of the shifter arm. Its a $10 part, just need a new one, case gasket, and a few tools, 20 min job tops and you can clean the oil screen while you got that side off as well. Also check the clutch adjustment before tearing into it just in case.
If your lucky its just the tip of the stopper arm broke off, in some really bad cases the boss that the bolt it pivots on is broken off the case, but if you can find all the pieces and they generally all just end up laying down in the side of the motor, you can get someone that can weld aluminum put it back in place, thats how i have fixed a couple.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....