Former prime minister having pressure put on hospital (5)
I believe the answer is:
heath
'former prime minister' is the definition.
(I've seen this before)
'pressure put on hospital' is the wordplay.
'pressure' becomes 'heat' (I've seen this before).
'put on' says to put letters next to each other.
'hospital' becomes 'h'.
'heat'+'h'='HEATH'
'having' acts as a link.
(Other definitions for heath that I've seen before include "blasted place?" , "Moor" , "Tract of open land" , "See 16" , "Sir Edward - , PM" .)