If there isn't any debris in the transmission area then the neutral valve is not operating.
This is as stated in the service manual:
When the clutch lever is released, the pump control bowl pushes the neutral valve open. When the mower is pulled backwards,oil is pumped out of the motor cylinder.
Anyway, a problem with Honda HRX transmissions.
If you are willing to take it apart and clean the check valve and replace the fluid you can do it yourself.
I don't remember Honda selling parts either.