A broken winded horse is, essentially, an animal that has a permanent respiratory condition that affects its usability. This is most commonly COPD or heaves, but the term is older - dating back to at least the 19th century. (Thus, you could probably use this term in a Medieval fantasy or historical without it looking like an anachronism).
In the past, horses with broken wind were often culled, especially if the condition was severe. They are now treated - but the condition is not curable. In many cases, horses with broken wind cannot be stabled for long periods, have to have their hay soaked and may need medication.