The subordinating clause of this sentence is... when we went to the Fourth of July picnic. Hope this helps!
Answer:
The dreariness of the speaker’s life away from Innisfree.
Explanation:
The lines 'While I stand on the roadway, or on the pavements grey, I hear it in the deep heart’s core' refer to a feeling of closeness to and remembrance of a place dear to the speaker’s heart. There is an implicit sense of removal, of physical distance, contrasted to an emotional proximity.
So we know it reflects his life away from the idyllic Innisfree. Futhermore, the general tone of the phrase, the depiction of the pavements' colour (rather a dull one), appear to suggest a certain general dreariness.