A cold clergyman, to be precise (8)
I believe the answer is:
accurate
'precise' is the definition.
(I know that precise can be written as accurate)
'a cold clergyman' is the wordplay.
'cold' becomes 'C' (eg on taps).
'clergyman' becomes 'curate' (assistant to a priest).
'a'+'c'+'curate'='ACCURATE'
'to be' acts as a link.
(Other definitions for accurate that I've seen before include "Exactly right" , "carefully and precisely" , "Correct, right" , "not deviating from the truth?" , "Care a cut is exact" .)