Take part assuming little left over (8)
I believe the answer is:
complete
'over' is the definition.
(over can mean complete or finished)
'take part assuming little left' is the wordplay.
'take part' becomes 'compete'.
'assuming' indicates putting letters inside.
'little' says to take the initial letters.
The first letter of 'left' is 'l'.
'compete' going around 'l' is 'COMPLETE'.
(Other definitions for complete that I've seen before include "whole,entire,uncut" , "Execute" , "Bring to a satisfactory end" , "Intact" , "accomplished" .)