With 'Agitato' indicating that that part has another 'mood' than the rest of the part, you could say it is unrelated of the rest in that way.
Tempo wise, since you have to make the melody sound 'agitated', the left hand is forced to play those notes quite fast to make that melody a whole in that mood.
Sometimes it helps to play the melody alone in the way you (or the composer) want it to sound like, to give you an impression how fast the accompanying notes are required to be played.
btw, a very nice nocturne.
Gyzzzmo