Hat that fellow nowadays regularly prizes going to church
I believe the answer is:
headpiece
'hat' is the definition.
(headpiece is a kind of hat)
'fellow nowadays regularly prizes going to church' is the wordplay.
'fellow' becomes 'he' (I've seen this in other clues**).
'nowadays' becomes 'ad'.
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
'going to' says to put letters next to each other.
'church' becomes 'ce' (abbreviation for Church of England).
The alternate letters of 'prizes' are 'pie'.
'he'+'ad'+'pie'+'ce'='HEADPIECE'
'that' acts as a link.
(Other definitions for headpiece that I've seen before include "hat" , "Something worn on top" , "Former intellect" , "decoration on top" .)