The only reference to Cowpits is the following sentence:
found in a description of the Parish of Inveresk in the Ordnance Gazetteer of Scotland: A Survey of Scottish Topography, Statistical, Biographical and Historical, edited by Francis H. Groome and originally published in parts by Thomas C. Jack, Grange Publishing Works, Edinburgh between 1882 and 1885, and available on the website Scottish People.
For references, see Parish of Inveresk
The National Library of Scotland has a map from the First Ordnance Survey One-Inch to the Mile Maps of Scotland 1856-1891 that shows this area in the middle of the 19th century. Many of the places cannot be found even on maps circa 1900.