Pressing on the old city chap (6)
I believe the answer is:
urgent
'pressing' is the definition.
(I know that pressing can be written as urgent)
'on the old city chap' is the wordplay.
'on' is a charade indicator (letters next to each other).
'the old city' becomes 'ur' (ancient city state).
'chap' becomes 'gent' (I've seen this in other clues).
'ur'+'gent' is 'URGENT'.
(Other definitions for urgent that I've seen before include "Demanding immediate attention" , "Requiring immediate action" , "High priority" , "Critical" , "Needing immediate attention, don't delay" .)