Organ grinder's heart eaten by bear (4)
I believe the answer is:
lung
'organ' is the definition.
(I know that Lung is a type of organ)
'grinder's heart eaten by bear' is the wordplay.
'heart' says to take the centre.
'eaten by' means one lot of letters goes inside another (inserted letters get eaten).
'bear' becomes 'lug' (both can mean to carry around).
The central letter of 'grinder' is 'n'.
'n' inserted within 'lug' is 'LUNG'.
(Other definitions for lung that I've seen before include "One of a pair of organs in the ribcage" , "Of sound mind" , "urban area" , "Breathing apparatus" , "Air bag?" .)