What is the name for a small piece of rock or metal that has broken off from a comet?

Not a meteorite unless it enters Earth's atmosphere. Pieces break off all the time, but only a very few enter our atmosphere. The vast majority will never become a meteorite.

Just as they say they meteor reaches both moon or earth's surface the question is do they find them flying/floating on moon b'coz there's absolutely no gravity there? IDK.

I love 💕 to collect rocks

This answer is Bravo Sierra. The fragments Must strike the ground.