Stiff found in church on street near the centre of Kilmarnock (6)
I believe the answer is:
starch
'stiff' is the definition.
(I've seen this before)
'church on street near the centre of kilmarnock' is the wordplay.
'church' becomes 'ch' (common abbreviation).
'on' says to put letters next to each other.
'street' becomes 'st'.
'near the' is a charade indicator (letters next to each other).
'centre of' indicates the central letters.
The middle of 'kilmarnock' is 'ar'.
'st'+'ar'='star'
'ch' after 'star' is 'STARCH'.
'found in' is the link.
(Other definitions for starch that I've seen before include "Potato, rice constituent; a stiffener" , "Stiffener in the laundry" , "rigid" , "Formal quality" , "Carbohydrate; stiffener" .)