Lack of interest, in a way, on the fourth of July (6)
I believe the answer is:
apathy
'lack of interest' is the definition.
(I've seen this before)
'a way on the fourth of july' is the wordplay.
'way' becomes 'path' (synonyms).
'on' is a charade indicator (letters next to each other) (in a down clue, letters appear on others).
'the fourth of july' becomes 'y' (4th letter of 'july').
'a'+'path'+'y'='APATHY'
'in' acts as a link.
(Other definitions for apathy that I've seen before include "Lack of emotion and interest" , "State of listlessness or disinterest" , "Unconcern" , "Torpor" , "Detachment" .)