I very much like the idea of this logo with the H as a jump, but the horse needs some work. Maybe use the pictures that I uploaded as a reference for how a horse ideally should jump. The horse's back should be arched upwards, and the neck downwards.
like the horse better, horse needs to jump a little higher over the 'H' so he's not touching the bar. Also, maybe make the horse a little shorter, not so much of his back and body showing. i like the flag idea but not as a circle , it makes me think of a western barn. One last thing, my theme colors are green and beige, so maybe the text can be changed to those colors.