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' is an anagram of 'PRIG'.
'with a' is the 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" .)