So Mary Frances Lockhart is the apparent link back to the people you named in your first post.
Looking for her in the index on FamilySearch and ScotlandsPeople came up with Mary Frame Lockhart, which (without purchasing the image) I felt was probably her. Frame could be a poor transcription of Frances. Her birth date is given as 25 Oct 1881. Is this correct - do you have her birth date at all to compare? Although I suspect not since you only gave a year of 1882.
The previous child listed in your first post is Ann with a birth year of 1881. Looking for her in the FamilySearch index, she is definitely the child of John Lockhart & Helen Moffatt and her birth date is 7 April 1881.
The only Mary Lockhart that I can see born in Hamilton in 1882 is shown in the ScotlandsPeople index but not FamilySearch. And there is no middle name of Frances
May I ask where the the birthplace of Hamilton came from? Is it from an official record?
The census
references or addresses (especially since searching we cannot be sure whether the family was in Scotland or England at any given census) where the family has been found would be useful to know so that we can look to see what other clues might be gleaned.
I'm having real trouble tracking where the Fernancy Jane reference has come from. I can see that the family of John & Helen (Moffat) moved from Scotland to England to India, back to England and then Scotland again from the 1881 census record, but the addition of this child in 1871 so soon after John in 1870 is puzzling, especially since I can't find any likely looking child on SP or FamilySearch. Can you enlighten us as to where you found this?
We all love puzzles, but I'm concerned that we may be following the wrong path here.
Nell