Posh doctor with anger giving offence (7)
I believe the answer is:
Here is my best explanation:
'offence' is the definition.
(I've seen this before)
'posh doctor with anger' is the subsidiary indication.
'posh' becomes 'u' (abbreviation for upper-class - opposite to non-u).
'doctor' becomes 'mb' (Bachelor of Medicine, Medicinae Baccalaureus).
'with' means one lot of letters go next to another.
'anger' becomes 'rage' (rage is a kind of anger**).
'giving' is the link.
(Other definitions for umbrage that I've seen before include "Resentment, offence", "Take this to show offence", "Suspicion of injury", "Feeling of anger caused by being offended", "Take this to feel offence", "Offence, annoyance", "Indignation", "Anger at having been offended", "feeling" and "It''s given or taken in offence".)