Offer has run to porridge (4)
I believe the answer is:
bird
'to porridge' is the definition.
(I've seen this before)
'offer has run' is the wordplay.
'offer' becomes 'bid' (bid is a kind of offer).
'has' indicates putting letters inside.
'run' becomes 'r' (cricket abbreviation).
'bid' going around 'r' is 'BIRD'.
(Other definitions for bird that I've seen before include "Egg-laying vertebrate" , "Nest builder?" , "Prison time (slang)" , "chat, perhaps?" , "Egg-laying creature" .)