Petition that doctors threaten, saving hospital (7)
I believe the answer is:
entreat
'petition' is the definition.
(thesaurus)
'doctors threaten saving hospital' is the wordplay.
'doctors' indicates an anagram.
'saving' indicates named letters should be taken away (saving can mean except).
'hospital' becomes 'h'.
'threaten' with 'h' taken out is 'treaten'.
'treaten' with letters rearranged gives 'ENTREAT'.
'that' acts as a link.
(Other definitions for entreat that I've seen before include "Ask (someone) anxiously" , "Beseech, implore" , "Beg earnestly, beseech" , "Pray" , "Beg earnestly or emotionally" .)