Yes, I do use the free mulch dropped off by arborists. I use this very chunky woody chips  around shrubs and as informal pathway material when it is dry but I have a pile that has been rotting for over a year and use the inner chips that are moist around woodland plants and rhododendrons. I do think you would be happy with the results if you layer an inch or two on top of the moss or fungus on your soil. You could also just turn the soil over to hide the growth as aerating the soil will help to discourage moss and fungi.  Don't worry too much as what you are seeing is most likely a natural growth from a wet winter and moss and soil fungi do not harm your plants. These soil covering colonies are simply opportunists that grow where nothing else cares to grow. A mulch of any type will tidy things up for you. Keep growing, Marianne Binetti