The intriguing thing here is the name of the old parish: Half Morton.
Once upon a time there was Morton parish (beware: there is yet another one of that name in Dumfriesshire, around Thornhill and Carronbridge).
Half of it was annexed to Canonbie parish in the east. What remained of the original was therefore - and with pedantic literalness - named Half Morton.
The former Half Morton parish kirk, built largely in the 1740s, is now a private house - with gravestones in the garden.