Start off having great influence, but score less than a hundred (6)
I believe the answer is:
eighty
'score less than a hundred' is the definition.
(a score can mean twenty)
'start off having great influence' is the wordplay.
'start off' means to remove the first letter (the start of the word is taken off).
'having great influence' becomes 'weighty' (weighty can mean influential or important).
'weighty' with its initial letter removed is 'EIGHTY'.
'but' is the link.
(Other definitions for eighty that I've seen before include "Several score" , "LXXX" , "Fourscore" , "score less than a hundred" , "XXC" .)