It hobbles around hospital with a slogan (10)
I believe the answer is:
shibboleth
'a slogan' is the definition.
('shibboleth' can be a synonym of 'slogan')
'it hobbles around hospital' is the wordplay.
'around' indicates an anagram.
'hospital' becomes 'h'.
'it'+'hobbles'='ithobbles'
'ithobbles' with letters rearranged gives 'shibbolet'.
'shibbolet'+'h'='SHIBBOLETH'
'with' is the link.
(Other definitions for shibboleth that I've seen before include "Catchword, slogan" , "test word" , "cultural custom" .)