Carry On Priests? Unlikely (7)
I believe the answer is:
persist
'carry on' is the definition.
(I've seen this before)
'priests? unlikely' is the wordplay.
'unlikely' indicates anagramming the letters.
'priests' anagrammed gives 'PERSIST'.
(Other definitions for persist that I've seen before include "Continue in the face of resistance" , "Continue firmly or obstinately" , "Continue in spite of difficulty" , "Continue to exist" , "Continue (in spite of difficulties)" .)