I use both and with good results, cow has some extra nutrients. Organic is good to be sure of avoiding aminopyralid, check that the horse hay did not come from a field sprayed with that awful weedkiller (google aminopyralid to learn more….).
The main thing is for manure to be well composted, about a year altogether, reasonably dark and crumbly, so that nutrients are stable, not water soluble, and any bedding such as straw is mostly broken down, to avoid it being mixed with salad! I would put some down before covering with plastic or you could fold sheets back and put it on top, then re-cover.