Lesson horses develop all kinds of defenses against beginning riders, with the long term goal of ALL of their possible riders giving up and letting said lesson pony live a life of idle luxury in the pasture.
When I end up on one of these burnt out lesson horses I approach it from the viewpoint that it is up to me to PROVE to the horse than I am a better rider than their usual riders. I often have to get creative. If one thing does not work after several rides I try something else.
One thing that has helped me lately with balking horses or horses that refuse to move forward freely is tapping them on the croup when that leg is supporting-pushing off. The correct moment to use this aid is when your hip rises, tap the top of the pony’s croup on that side. When I tap on the croup I keep my reins in the other hand so I do not give an accidental hand aid. This action is a tap, when I tested it on my bare leg it just smarted a little bit. This is an aid, not a punishment, I am trying to get the horse to push off of that hind leg a with a little bit more push. I do it mostly at a walk because right now I am too uncoordinated to do it at a trot, but after practicing it at a walk you might be able to do it when said pony just drags at the trot.
I repeat, this is just a tapping motion, tap each time the croup rises on that side, and continue until the pony moves a little faster. DO NOT do this hard, if it is done too hard the rider might incite a sudden charge forward, a runaway, or the horse can kick. Be sure to praise the pony when the pony goes faster like you want it to. If the pony goes too fast at first let the pony move for a few strides before you tell him to slow down to a more reasonable speed.
This has worked for me with dead sided lesson horses, and on horses that obviously never learned what a leg signal at the girth means. It is very, very, very important not to accidentally give a rein aid at the same time.
Good luck with the pony!