Since you MUST ride your mule with the sore on him. You are going to have to do something to PREVENT RUBBING. [which is obviously how the sore became.
What iwould do is use some antibacterial stuff to disinfect the wound. Then I would lather it in bag balm, then cover it up with a THICK rag, then cinch him up with the rag under the cinch, you can to very hard riding, but you'll be able to train some.
Also, you're going to have to make sure the rag stays intact. So what I would do is use duck tape to tape the rag to the mule, not the cinch.
I know this isn't the best method, but riding with something like this isn't a very good method and calls for odd measures, per se.