What type of hackamore is she in?
With this mare being owned by a beginner it's very possible she has never had to ever do more than walk with a rider aboard. I would address this as a horse that is very green more than one just being a brat.
How is she moving off pressure on the ground? If you're nervous about her bucking that is the first place to start. Put her in the round pen and cluck and point the direction you want her to go ( or whatever you want your verbal trot cue to be) followed by swinging the whip, getting progressively closer to her with it ( whacking her on the bum if needed until she trots off. As soon as she trots off away from you, quit putting any pressure on her. Do not cluck, move your ship etc. Just follow her hip on a small circle in the roundpen. If she breaks to a walk fine, all the more opportunity to teach her. Again point, cluck then use the whip. If she gets pissy about moving forward really push her forward until she's with the program. You want her to be happy to move off on the ground first.
If she only trots one step at a time that is A OK. At this point you're only teaching her the cue TO trot. Once she is good about picking up a trot when you ask THEN you can go about keeping her in a trot. Every time she breaks to a walk immediately push her back up into a trot.
Repeat with the lope. In doing this you've got her understanding go without you being in danger and also got her sensitized to a whip. The next step would be to desensitize to it, so she doesn't think any time a whip moves it means go.
I'd also teach her to flex before you ride her next.
When you do get to riding the course of action is pretty darn simple and much safer for you now. Flex her head then slide your leg back to ask her hindquarters to move off, don't kick, just a light squeeze.
Many times kicking a horse is going to just cause them to sull up even more, raising up from the pressure instead of going forward, once they're already no longer thinking forward at all then it goes from a hump to a buck, or a rear. Either way it ain't good.
So squeeze with your leg then take your rein or whip ( for less typing i'll be saying rein) and just very casually tap her bum, getting progressivly harder until her hind end untracks. In this position you're much better off if she does decide to spook from the rein. If she does no big deal, just keep repeating this until she's moving off but not in a panic. Repeat on both sides.
Once that is down then you can get a bit more firm. Tap her a bit harder on the butt as she's disengaging and keep her moving, when you feel her front end start to move let her head out and let her walk on. You want that hind end to have some energy to it when you let her walk out. If she stops from a walk, bend her around and get that hind end moving like you did before.
You also want to make sure you can bend her to a stop so if things to go astray you have a stop.
From here, use your trot verbal cue from the ground, squeeze lightly with your legs then enforce it with your rein. If you are say going clockwise have your right rein slightly shorter so you can bend her down easily and keep a hold of that saddle horn if you want. Your left hand is then free to use your rein.
If she bucks you don't need to worry about cowboying up and riding through it if you're nervous, the important thing is that action is taken quickly. Bend her down and get after her, making that hind end move around pretty snappy. Then calmly ask her to move out again. As long as you are doing something to make that buck a difficult thing for her you are fine.
However, once she is happy to move forward on the ground you'll find she'll be much much easier and willing to go under saddle too.