Man from Edinburgh, say, to rush, avoiding a round (4)
I believe the answer is:
scot
'man from edinburgh say' is the definition.
(Edinburgh is in Scotland)
'rush avoiding a round' is the wordplay.
'rush' becomes 'scoot' (to scoot is to rush away from somewhere).
'avoiding' is a deletion indicator.
'a round' becomes 'o' (round can mean a round object).
'scoot' with 'o' removed is 'SCOT'.
'to' acts as a link.
(Other definitions for scot that I've seen before include "Ancient tax" , "Glaswegian, say" , "Glaswegian perhaps" , "Payment" , "Eg Aberdonian" .)