Pick up body part by hospital (4)
I believe the answer is:
hear
'pick up' is the definition.
(hearing is a kind of picking up)
'body part by hospital' is the wordplay.
'body part' becomes 'ear' (ear is a kind of body part).
'by' means one lot of letters go next to another.
'hospital' becomes 'h'.
'ear' put after 'h' is 'HEAR'.
(Other definitions for hear that I've seen before include "receive information" , "Hark" , "Be informed of" , "Perceive aurally" , "Get wind of" .)