Heavy (turf)
Wettest possible condition of a turf course; not usually found in North America.  

Helmet
A lightweight fiberglass cap worn by riders to prevent head injuries. It is required equipment that is not considered part of a jockey's riding weight.  

High Weight
Highest weight assigned or carried in a race.  

Homebred
A horse bred by its owner.  

Horse
When reference is made to sex, a "horse" is an ungelded male five-years-old or older.  

Hot Walker
Person who walks horses to cool them out after workout or races.  

Hung
A horse that does not advance its position in a race when called upon by its jockey is said to have hung.