Our establishment’s trendy, in rear street with headgear British and European (3,6,4,2)

'our establishment's' is the definition.
('the powers that be' can mean the establishment)

'trendy in rear street with headgear british and european' is the wordplay.
I cannot quite see how this works, but
'trendy' could be 'hep' (I've seen this in another clue) and 'hep' is present in the answer.
'rear' could be 'tower' (towering is a kind of rearing) and 'tower' is located in the remaining letters.
'street' could be 'st' (abbreviation) and 'st' is found in the answer.
'headgear' could be 'hat' (hat is a kind of headgear) and 'hat' is found within the answer.
'british' could be 'b' (abbreviation e.g. in 'BBC') and 'b' is located in the answer.
'european' could be 'e' (abbreviation e.g. EU) and 'e' is found within the answer.
No letters remain.
This explanation may well be incorrect...