Singer that’s forgotten organist’s central piece (5)
I believe the answer is:
tenor
'singer' is the definition.
(I know that tenor is a type of singer)
'forgotten organist's central piece' is the wordplay.
'piece' indicates a hidden word.
'TENOR' can be found hidden inside 'forgotten organists central'.
'that's' acts as a link.
(Other definitions for tenor that I've seen before include "Higher male voice" , "High voice - low bell" , "character" , "Vocal range" , "General meaning of a male voice" .)