Keen to embrace that woman in bright red (6)
I believe the answer is:
cherry
'bright red' is the definition.
(I've seen this before)
'keen to embrace that woman' is the wordplay.
'keen' becomes 'cry' (I've seen this in other clues).
'to embrace' indicates putting letters inside.
'that woman' becomes 'her' (I've seen this before**).
'cry' going around 'her' is 'CHERRY'.
'in' acts as a link.
(Other definitions for cherry that I've seen before include "Stone fruit - shade of red" , "'Tree, fruit and colour (6)'" , "Bright red" , "Deep red like small round fruit" , "Tree; red fruit" .)