Constituent very nearly claims overtime (5)
I believe the answer is:
voter
'constituent' is the definition.
(synonyms)
'very nearly claims overtime' is the wordplay.
'nearly' means to remove the last letter.
'claims' means one lot of letters goes inside another.
'overtime' becomes 'OT' (O and T are abbreviations for over and time).
'very' with its last letter taken off is 'ver'.
'ver' placed around 'ot' is 'VOTER'.
(Other definitions for voter that I've seen before include "Elector" , "Lively dances (esp. Irish)" , "One of the selectors" , "Constituent" , "supporter of politician" .)