Avoid woman who was tempted to grab publicity (5)
I believe the answer is:
evade
'avoid woman who' is the definition.
'evade' can be an answer for 'avoid' (I've seen this before). I'm not sure about the remainder of the definition.
'tempted to grab publicity' is the wordplay.
'tempted' becomes 'eve' (I can't justify this - if you can you should give a lot more credence to this answer).
'to grab' means one lot of letters goes inside another.
'publicity' becomes 'ad' (advert).
'eve' enclosing 'ad' is 'EVADE'.
'was' acts as a link.
Can you help me to learn more?
(Other definitions for evade that I've seen before include "skirt" , "Escape (esp. by guile)" , "bypass" , "Shun, escape" , "avoid answering" .)