The word mush, meaning something soft and sloppy and probably not in its best state, comes via the American colonies, as a variant of mash; at that time, in the late 1600’s it had a specific meaning of a type of porridge. This came to mean anything soft, and thick and wet, a mixture of something which had been mashed and pounded to a pulp, and that has been found in a written form in the 1820’s. There is another mush, what is supposedly said to huskies when they are pulling a sledge; maybe that came from the French ‘marcher’, but really who knows!
There is another mush though; mush meaning mate or pal, but used to address someone whose name is unknown. it can seem almost derogatory, and is often used with ‘oi!’ It can also mean a face, again in a derogatory sense ‘I smacked him in the mush‘, for example! These two uses which may have the same origin, date back to the nineteenth century and maybe Romany in origin.
Here’s an interesting article about it: