1.) First I'm assuming she plants her feet when she's being lead with or WITHOUT a rider, so start trsaining withOUT a rider. Standing on left side of pony's shoulder holding lead in right hand (and a LONG whip in left hand), wlak forward. If when you step forward she hesitates or does not step forward, tap pony with whip on butt (middle by the tail). If she even takes 1 step forward nicely tell her good girl. If she leaps forward just hold onto her and let lead line restrict her forward progress, and keep walking.
The idea here is to condition her to walk forward immediately upon request by the handler (pony should mimic handler feet, as you step forward she should step forward).
If instead of forward movement pony kicks out at whip, say a LOUD "NO" and immediately smack pony harder with whip. Let it sink in a minute then repeat asking nicely for pony to walk forward.
2.) When pony walks forward promptly as handler asks then you can repeat this with a rider. Rider should have something to hold onto in case pony bolts forward.
And FYI - there is a condition called kissing spines which, when you add the weight of a rider onto a horses back, can cause the horse pain - thus that could be a source of pony's "occassional" bad manners, so it MAY be pain related, but I would start with the above "instruction" to install better manners on the pony. Never forget that YOU are herd leader and pony is expected to listen to what you are telling her, then make certain pony "understands" what you're asking her to do (before any punishment/negative reinforcement).