Absence of a street name with the final double-S (8)

'absence' is the definition.
(awayness is a kind of absence)

'a street name with the final double-s' is the wordplay.
'street' becomes 'way' (I've seen this before).
'name' becomes 'n' (common abbreviation e.g. on forms).
'with' says to put letters next to each other.
'final' indicates one should take the final letters.
'double' shows that letters should be duplicated.
The final letter of 'the' is 'e'.
's' doubled is 'ss'.
'a'+'way'+'n'+'e'+'ss'='AWAYNESS'

'of' is the link.