I believe the hose you mention is for the headlamp washer system, the small wiper arms on the headlights.
If you start the car and switch on the headlights, then activate the windshield washer, the headlights are sprayed with washer solvent and the small wipers run for a few seconds. Only if the headlights are on though, and only for a few seconds. They don't run continuosly. Hope this helps.