One of the most common pitfalls for gymnasts on bars is the free-hip (or clear-hip) circle. In the Level 6 and 7 routines, a minimum height must be achieved to receive full credit for the skill, and a large deduction is taken when it's not high enough. Ideally, for full credit and maybe even leaving a good impression on the judges, you would be best off trying to hit a free-hip to handstand.
Many gymnasts think a free-hip is like a back hip circle, except you just open up at the end; you just go around the bar and open your shoulders at the right time. Well, it's not quite that simple. If you've worked free-hips for long enough, you'll know that whipping your body around the bar isn't the only factor in achieving a good free-hip. If you think about it, will it be easy to open up to handstand by simply rotating around the bar in a perfect circle? It's all physics! If the center of gravity of your body is moving in a perfect circle, it will want to continue moving in that same circle.
So, in order to do a good free-hip, your center of gravity will need to move in more of a vertical oval. Think of it this way: in order for an object to have upward momentum, it first needs some downward energy. If you want to throw a ball straight up into the air, you pull your arm down first before throwing upward, right? Well, a similar technique is required for you to get your body to handstand.
To apply this, you need to pull down on the bar (your momentum going toward the floor) after you cast. When you hit the bottom of the swing, and are upside down, the bar should be near the middle of your thighs. Here's where it's different from a back-hip circle; you have more downward momentum, and the bar is away from your hips, hence the name "free-hip." But be careful. If you pull the bar too far on your legs, you might fling off the bars as you're coming up.
Once you get this, the technique to shoot up to handstand is somewhat more complicated than it looks, and it involves several things going on at once. Use that downward energy explained above and transfer it to upward momentum by shooting your legs a little short of vertical (as if you were to fall back over if you didn't quite make it to handstand -- but you won't if you have enough momentum). You should flip your wrists around the bar as quickly as possible to provide a good support to get up to a handstand. Once your wrists have flipped, it makes it easier to open your shoulders.
Remember to stay tight. If you're loose, you'll lose a lot of the potential energy you can get, and it'll only make it harder for you.
Free-hips can be a strength move if you're not strong enough in your arms and shoulders to shoot to handstand and hold it. While there isn't one specific exercise to condition those essential muscles, there are several you can do that will make this easier. You can do "kiss-the-wall handstands," shoulder shrugs, hollow-body push-ups, and handstand push-ups.
Put all your effort into it, and follow any other advice your coach might give you. Good luck!