Posh doctor with anger giving offence (7)
I believe the answer is:
umbrage
'offence' is the definition.
(I've seen this before)
'posh doctor with anger' is the wordplay.
'posh' becomes 'u' (abbreviation for upper-class - opposite to non-u).
'doctor' becomes 'mb' (Bachelor of Medicine, Medicinae Baccalaureus).
'with' says to put letters next to each other.
'anger' becomes 'rage' (rage is a kind of anger).
'u'+'mb'+'rage'='UMBRAGE'
'giving' is the link.
(Other definitions for umbrage that I've seen before include "Resentment, offence" , "Displeasure" , "Feeling of anger caused by being offended" , "Offence, annoyance" , "Offence that's usually taken" .)