Chance of promotion within popular hospital department (11)
I believe the answer is:
inadvertent
'chance' is the definition.
(thesaurus)
'promotion within popular hospital department' is the wordplay.
'promotion' becomes 'advert' (advert is a kind of promotion).
'within' means one lot of letters goes inside another.
'popular' becomes 'in' ('in' can mean trendy).
'hospital department' becomes 'ENT' (abbreviation for Ear, Nose and Throat department).
'in'+'ent'='inent'
'advert' inserted inside 'inent' is 'INADVERTENT'.
'of' acts as a link.
(Other definitions for inadvertent that I've seen before include "unguarded" , "Careless" , "Accidental" , "Unintentional" , "Unthinking" .)