Go on, sir. Chuck the pest out! (7)
I believe the answer is:
persist
'go on' is the definition.
(persisting is a kind of going on)
'sir chuck the pest out' is the wordplay.
'chuck' indicates anagramming the letters (I've seen 'chucked' mean this (letters chucked around)).
'out' indicates putting letters inside.
'sir' with letters rearranged gives 'rsi'.
'rsi' put into 'pest' is 'PERSIST'.
(Other definitions for persist that I've seen before include "Continue to exist" , "Continue (in spite of difficulties)" , "Continue firmly or obstinately" , "Pursue course doggedly" , "Persevere, hang on" .)