Terribly ill around hospital department, showing a pulse (6)
I believe the answer is:
lentil
'a pulse' is the definition.
(type of pulse)
'terribly ill around hospital department' is the wordplay.
'terribly' is an anagram indicator.
'around' means one lot of letters goes inside another.
'hospital department' becomes 'ENT' (Ear, Nose and Throat department).
'ill' with letters rearranged gives 'lil'.
'lil' going around 'ent' is 'LENTIL'.
'showing' is the link.
(Other definitions for lentil that I've seen before include "bit of food" , "Type of soup" , "Plant - pulse" , "Kind of bean plant used in soup" , "High-protein pulse" .)