York Kings Manor is situated to the left of the Art Gallery and also at the rear of the Museum Gardens. Its existence began in 1270 as the Abbot's lodging who obviously was the Abbot of St Mary's Abbey behind it. However, when first built it was much smaller and has been extended twice, firstly in 1480 and also in 1610.
The Kings Manor survived the Dissolution of the Monasteries unlike the abbey itself because quite simply, Henry VIII admired the building.
When the Abbey dissolved, King Henry VIII instructed that it be the seat of the Council of the North until 1641 when the council was abolished. Today it serves as part of the University of York and you can study archeology here. In addition, the building is open to the public.
Along the left hand side of the grounds you can walk to the Museum Gardens along a public right of way.