Puritan with a shaky grip (4)
I believe the answer is:
prig
'puritan' is the definition.
(both can mean a prudish person)
'shaky grip' is the wordplay.
'shaky' indicates an anagram.
'grip' with letters rearranged gives 'PRIG'.
'with a' acts as a link.
(Other definitions for prig that I've seen before include "Annoyingly arrogant person" , "Sanctimonious person" , "humbug" , "Over-fussy and self-righteous type" , "Stuffed shirt" .)