Air Carrier flight crew reported on short final to DSM ATC cleared an aircraft to cross the landing runway; the flight crew executed a go-around to avoid the conflict and was vectored back to the airport resulting in a successful landing.

Narrative

Approach vectored us for an ILS RWY 13.Called field in sight and subsequently Approach cleared us for the visual to RWY 13.While approaching the Final Approach Fix the First Officer queried Approach for a hand off to tower.Hand off to tower just prior to CLIVE at 2;600' MSL.First Officer contacted tower and tower cleared us for landing.Tower then started to speak with another aircraft that was taxiing.The clearance was for that aircraft to taxi to a specific location but to Hold Short of RWY 13 on Papa.There was no response from the aircraft taxiing.Tower repeated the clearance and again stated to hold short of RWY 13 - still no reply from the taxiing aircraft.The First Officer and I looked at each other acknowledging this could be an issue.Tower then changed the clearance and approved the taxiing aircraft to cross RWY 13.First Officer queried tower and asked about our landing clearance.Tower replied with a comment that we were previously cleared to land.Approaching 300' AFL we watched the taxiing aircraft begin to cross RWY 13 on Papa taxiway.I felt this was creating an unsafe situation and called for a pilot initiated Go Around.Performed a Go Around and subsequently was vectored back for an ILS RWY 13.Uneventful landing and taxied to gate.Tower was working ground and tower utilizing 2 different frequencies. Tower cleared an aircraft to cross the active runway while we were on short final causing an unsafe situation and unacceptable clearance between the landing aircraft and the taxiing aircraft. A late hand-off from Approach Control to Tower may also have contributed to the situation.Consider dedicated controllers for ground and tower. At many smaller airports; one controller performs multiple tasks and this can lead to a reduction in safety.