So in your back handspring, the squat is really important. If you feel like you're undercutting it or piking down, it might have something to do with the take-off. When you're going into your squat, make sure that your knees are in line with your feet (not in front of them). The backwards motion starts at the very beginning, so you should feel like you're sitting back in a chair. That way, when you push off with your feet you'll get a nice backwards motion and a good arch position over the top.
In any flip skills (tuck, pike, layout) the take-off is also VERY important. The main thing is that you need to get a good set before you tuck (or pike, or flip in general lol). You can practice getting a big set out of your roundoff back handspring, as long as you have a coach there to support your back. If you're tucking immediately out of the rebound or throwing your head back, your form will be sloppy and incorrect. Same applies to twisting moves; you still need to set before you flip and before you twist. Twisting right away is a mistake.
Other than that, obviously make sure you're getting a good sprint and a powerful hurdle into the roundoff before any tumbling pass. Keep your body tight and you should be good to go!