HPN Controller described a go around event when a taxiing aircraft failed to report clear of the landing runway; the situation complicated by very low surface visibility.

Narrative

I kept the Tower open with all the positions combined. Several itinerant aircraft landed; but they did not have any parking at the main terminal. I staged these aircraft at several locations on the field. In addition; I was clearing aircraft to reposition for overnight parking; clearing GA aircraft to their FBOs and issuing IFR clearances to aircraft looking to depart HPN. I was coordinating with Westchester County Operations about the Tower visibility; gate availability; and other aspects of the operation and coordinating with N90/LGA regarding PIREPS and Tower visibility. Rapidly decreasing visibility conditions; down to 1/4 SM; occurred. At the time of the event; Aircraft Z was holding short of Taxiway Foxtrot on Taxiway Lima. Aircraft X was holding short of Runway 11 on Taxiway Lima. Aircraft Y was on a 7 mile final and cleared to land when Aircraft X was on the opposite side of the field from the main terminal; cleared to turn right onto Runway 11; cross Runway 16; then turn left onto Taxiway Alpha; and taxi to the main terminal. Aircraft X was advised of the rapidly decreasing Tower visibility and instructed to advise clear of Runway 16. Aircraft Y's landing clearance was cancelled. Aircraft Y was told that they would be advised when Runway 6 was clear. I waited until Aircraft Y was on a 3 mile final but had not received verification of Aircraft X's position. At that moment; Aircraft X asked me where they were supposed to clear the runway. I asked Aircraft X to confirm their current position; and they replied 'on Runway 16.' Aircraft Y was on a 2 mile final. I immediately cancelled Aircraft Y's approach clearance and issued 'Go Around' procedures. Aircraft X advised they were clearing Runway 16 at Taxiway Bravo. Recommendations; hold at least two CPCs on overtime and pay more attention to pilot read backs for their accuracy.