There is an estate called Corehouse near Lanark, though it's actually in Lesmahagow parish. I've looked on the IGI but there are no births/baptisms of an Alexander Thom(p)son in either Lanark or Lesmahagow between the dates you mentioned.
The 1851 census shows an Alexander Thomson, aged 1, born Lesmahagow living with his parents (father is also Alexander) and siblings in Kirkfieldbank, which is a village near Corehouse in Lesmahagow parish. However the father is a cotton handloom weaver, not a farmer. Do you know Alexander's mother's name? - the mother in the census seems to be called Rose or Rosa. (The family is also in the 1861 census, still in Kirkfieldbank, and the father still a weaver).
Lizci