Platthorn was just south-east of the Kikrton of East Kilbride. Its land has been swallowed up by the new town of East Kilbride. See
http://maps.nls.uk/geo/explore/#zoom=16&lat=55.76394&lon=-4.16708&layers=5 - use the slider on the left to compare this old map with the modern satellite view.
The farm house and buildings were inside the angle made by the junction of what are now Platthorn Road and Platthorn Drive, east of the former and south of the latter.
My great-great-grandparents Alexander Scott and Margaret Aitkenhead lived at Platthorn, and their three eldest children were born there in 1838, 1840 and 1841. I gather, however, that Alexander was not a successful farmer, and they had left Platthorn by 1843, when their fourth child was born. In 1841 there was one more house at Platthorn.
Later on, Alexander's niece Margaret Scott, who had married James Paterson, lived at Platthorn. Five of their six children were born there between 1856 and 1865 but they had moved away by the time their youngest was born in 1869. I have the details of their family from the 1861 census, but not of any other households nearby.
In 1881 there were three households at Platthorn: George Lyle, farmer of 102 acres, with wife, three children and a servant; Mary Strang and her son James Young; and Andrew Paterson with wife and servant.
I don't have information about who the tenants were in the censuses in between.
Hope this helps.