What to do with apple mush from making cider??

I haven't attempted this myself before, but I imagine that you would be able to use it like you would apple sauce in any baking recipe like this one: https://food52.com/recipes/8646-applesauce-cake-with-caramel-glaze

That apple pomace really fires up the compost heap.

I have an old hand cranked apple press and I usually feed the mush to my cows and horses, but most people don't have those. :-)

If it's actually apple pulp you might be able to push it through a sieve and then cook it down into apple butter. Peels and cores could be cooked with more liquid to make an apple juice for jelly and if there's not enough apple flavor you could add cranberries or elderberries for more flavor. If you put it through a sieve the resulting mush will be applesauce, which can be used in the traditional manner.

some baking recipes allow you to use applesauce instead of the oil or fat.

Bake it into muffins or quick bread? Or good ole' compost :-)

Make applesauce or pectin, http://www.instructables.com/id/Using-Apple-Leftovers-from-Juicing-Pectin-Appl/