I will not appear in most significant piece (3)
I believe the answer is:
'piece' is the definition.
(man can mean a chess piece)
I can't explain the remainder of the clue.
Can you help me to learn more?
(Other definitions for man that I've seen before include "-- Friday" , "The most destructive and polluting animal" , "Fellow" , "Isle of -; chess piece" , "Geezer" .)