My daughter’s pony does a similar thing. Actually, this is kind of embarrassing, but he starts every ride with two coughs and a fart (sorry if it’s gross, but it’s true!). Her trainer has never seemed concerned about it, and he’s done it his whole life (23 years now!), so for him it’s normal. But if your horse’s cough/stretch is really new for him, that might be a different story. Good luck!