Escape publicity during the night before... (5)
I believe the answer is:
evade
'escape' is the definition.
(I know that evade is a more specific form of the action escape)
'publicity during the night before' is the wordplay.
'publicity' becomes 'ad' (short for advert).
'during' indicates putting letters inside.
'the night before' becomes 'eve' (eve can mean the night before something).
'ad' inserted within 'eve' is 'EVADE'.
(Other definitions for evade that I've seen before include "Illegally escape (taxes)" , "avoid answering" , "Dodge - prevaricate" , "Keep away from" , "Escape paying (tax)" .)