Dodgy priests remain (7)
I believe the answer is:
persist
'remain' is the definition.
('persist' can be a synonym of 'remain')
'dodgy priests' is the wordplay.
'dodgy' is an anagram indicator.
'priests' with letters rearranged gives 'PERSIST'.
(Other definitions for persist that I've seen before include "Prevail" , "Pursue course doggedly" , "Stripes (anag)" , "Continue despite opposition" , "Show doggedness" .)