if you cut it you can damage the channel paths that are leading to healthy sections, if its a case of air exposure don't treat it like you would a coral with an infection. treat it more like a dish sponge, keep the sponge submerged at all times, lift it and wave it back and forth like a pinwheel to see if any bubbles come out, if it is a really fine pore sponge you can gently squeeze it (gently being key here) and release presure several times to help it purge the air pockets, you can cut away dead tissue but try not to cut into living tissue, this can be very very tough especially with siliceous sponges as they arent easy to cut. (watch out for silica splinters, wear gloves) they dont dissolve like a calcerous one will. if your tank is young, or of very small water volume you may be safer discarding the dying sponge so it does not crash your tank some sponges are extremly noxious when dying.
unfortunatly one good sized sponge is capable of crashing a new or small system, 40$ in the trash is better than everything else combined.
sorry for your luck, next time you purchase a sponge INSIST that they do not remove it from the water at anytime and do not let them leave any air in the bag for the trip home.if they do either refuse to buy it.
hope this helps