Church has the edge in felony (5)
I believe the answer is:
crime
'felony' is the definition.
(I've seen this in another clue)
'church has the edge' is the wordplay.
'church' becomes 'CE' (abbreviation for Church of England).
'has' means one lot of letters goes inside another.
'the edge' becomes 'rim' (rim is a kind of edge).
'ce' enclosing 'rim' is 'CRIME'.
'in' is the link.
(Other definitions for crime that I've seen before include "Action punishable by law" , "Legal offence" , "Misdeed" , "Murder, perhaps" , "Illegal deed" .)