Your "Horse cover thingy" I'm going to assume is a rug of some sort, but just to be sure here is a rug, AKA stable sheet.

They come in varieing degrees of warmth and thickness, but their basic function is to keep the horse clean and warm in the stall.
From Horse-Canada.com (how to hot towel):
One winter (or anytime) trick that works well to remove dirt without bathing is hot towelling. Get several old towels, preferably white or light coloured. Wet one at a time in a bucket of very hot water and wring it out thoroughly and quickly (so it doesn't have time to cool off). Place the hot towel on one part of the horse's body at a time. The heat will draw out the dirt. Continue with one section at a time. Finish by rubbing with a dry towel and brushing.