The white mushrooms are very wierd at first and theyre hard to figure out but I'll tell you what they're for. Never use physical attacks on them- NEVER- that will make them disappear instantly. You've probably noticed that white myshrooms do different motions- whenever they do these motions use a certain magic spell on them depending on what they are doing. If they fall on the ground use cure, if they shiver use fire, if they wave themselves use blizzard ,if a light shines over its head use lightning, etc. (i dont remember the other ones). If you do the spell right and within a certain amount of time you'll hear a pinging sound and the mushroom will jump in excitement. If you do three successful spells on it it will disappear and drop items( usualy ones that you use for synthesizing) and the items it drops depends on what spells you use. If you do three of the same spell on it (you can wait for it to do the same motion again) it will drop an Arts item (if oyu use fira three times you get a Fire Arts) and if you get an Arts item for every spell you can trade them into Merlin for a new keyblade. Also, when you use 3 of the same spell on it you usually will get a mystery goo which you need to synthesize many items incluing the Ultimare Weapon. I find white mushrooms the most in the campsite in Tarzan's jungle.

shiver>fire
wave their arms>blizzard
orb over their head>thunder
lying on ground>cure
spinning>aero
floating>gravity
frozen>stop