Carry On publicity with a scary woman (8)
I believe the answer is:
progress
'carry on' is the definition.
(to progress is to carry on or advance)
'publicity with a scary woman' is the wordplay.
'publicity' becomes 'pr' (abbreviation for public relations).
'with' says to put letters next to each other.
'a scary woman' becomes 'ogress' (female ogre).
'pr'+'ogress'='PROGRESS'
(Other definitions for progress that I've seen before include "Go forward" , "Move forward" , "Advance - movement" , "Go on - journey" , "Forward movement towards a destination or goal" .)