Street initially coned off, to a degree (6)
I believe the answer is:
second
'degree' is the definition.
(thesaurus)
'street initially coned off' is the wordplay.
'initially' suggests taking the first letters.
'off' indicates an anagram.
The initial letter of 'street' is 's'.
'coned' anagrammed gives 'econd'.
's'+'econd'='SECOND'
'to a' is the link.
(Other definitions for second that I've seen before include "tick" , "Short time; attendant at a duel" , "Unsurpassed" , "Endorse, back" , "Time for a supporter" .)