Dodge first woman with publicity (5)
I believe the answer is:
evade
'dodge' is the definition.
(I know that dodge can be written as evade)
'first woman with publicity' is the wordplay.
'first woman' becomes 'eve' (first woman in the bible).
'with' indicates putting letters inside.
'publicity' becomes 'ad' (short for advertisement).
'eve' placed around 'ad' is 'EVADE'.
(Other definitions for evade that I've seen before include "Sidestep" , "Escape (something)" , "Dodge or escape" , "Escape (esp. by guile)" , "Manage to avoid" .)