Hesitate to make fine change (6)
I believe the answer is:
falter
'hesitate' is the definition.
(I know that falter is a more specific form of the action hesitate)
'fine change' is the wordplay.
'fine' becomes 'f' (abbreviation).
'change' becomes 'alter' (synonyms).
'f'+'alter'='FALTER'
'to make' acts as a link.
(Other definitions for falter that I've seen before include "Hesitate, begin to give way" , "Lose momentum" , "Be hesitant" , "Hesitate, be uncertain" , "Hesitate, waver" .)