Sleep on right side of essentially fussy miss (4)
I believe the answer is:
skip
'miss' is the definition.
(to skip something is to miss it out)
'sleep on right side of essentially fussy' is the wordplay.
'sleep' becomes 'kip' (synonyms).
'on right side of' says to put letters next to each other (some letters go to the right of others).
'essentially' says to take the centre.
The central letter of 'fussy' is 's'.
'kip' put after 's' is 'SKIP'.
(Other definitions for skip that I've seen before include "Pass over, omit" , "rubbish to go in it" , "Rubbish container - miss - jump" , "Omit - refuse container" , "Container used to collect and dump waste" .)