Noah almost traps rook for woman (4)
I believe the answer is:
nora
'woman' is the definition.
(woman's name)
'noah almost traps rook' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'traps' indicates putting letters inside.
'rook' becomes 'R' (chess abbreviation).
'noah' with its final letter taken off is 'noa'.
'noa' placed around 'r' is 'NORA'.
'for' is the link.
(Other definitions for nora that I've seen before include "-- Batty (Last of the Summer Wine)" , "Woman" , "Girl's name" .)