I sing all the time so I can breath (LOL) I used to do it as well while riding and playing tennis but when you sing or hum it's so much better.
I also will sing something if I need to.
Pick a favorite song u like and sing it in ur head.
When walking the course pick specific points where you will be breathing. Before starting what might also help is to stop the horse and take a deep breath. This helps the horse relax too and he/she might also take a breath and this helps in the overall relaxation.
This will also makes you stiff in the ride! Try not to open your mouth. And practice at home with breathing from your noses!
I used to have this problem lol. Sing (or hum) the alphabet or your song!! That way you're constantly breathing