@Susan J-D
Have you considered that Home Farm and Coney Garth may be one and the same thing? The name “Home Farm” is traditionally used to refer to any farm that is part of the manor house’s demesne – often the farm lying closest to the manor house. These were often farmed by a farm manager, hired by the lord of the manor, to cater for the needs of his household while other land was rented out to tenant farmers.
Looking at this map from 1859 you can see that Coney Garth (centre), located between Ashington and Pegswood, is a considerable size and just a stone’s throw from Bothal Castle (bottom left) and also part of the Bothal Demesne.
Of course, your relative may not have worked for the lord of the manor. Much depends upon when your relatives lived there. The name Home Farm can live on for centuries after being used as such.