Dan's out West with a woman (4)
I believe the answer is:
dawn
'woman' is the definition.
'dan's out west' is the wordplay.
'out' indicates putting letters inside.
'west' becomes 'W' (abbreviation).
'dan' placed around 'w' is 'DAWN'.
'with a' acts as a link.
(Other definitions for dawn that I've seen before include "Light in East" , "Aurora was its goddess" , "fresh start" , "Early morning sing-song?" , "Begin to appear" .)