Chaplain's flat rate, having no heart (5)
I believe the answer is:
padre
'chaplain's' is the definition.
(padre is a kind of chaplain)
'flat rate having no heart' is the wordplay.
'flat' becomes 'pad' (person's home).
'having no heart' says to hollow out the word (remove centre letters).
'rate' with its middle taken out is 're'.
'pad'+'re'='PADRE'
(Other definitions for padre that I've seen before include "Sky pilot (military slang)" , "Minister" , "Army chaplain" , "Army serviceman?" , "Services' chaplain" .)