This may or may not be helpful, but in the 1841 census in Bothwell, at No 20 Carnbroe Iron Works, is a household consisting of
James Kennedy, 29, iron miner
Mary Kennedy, 24
George Kennedy, 1
John Kennedy, 18, iron miner
Bridget Kennedy, 62
Mike McSorley, 30, smelting furnace filler
James Burns, 25, smelting furnace filler
John Burns, 28 labourer
Francis Collins, 30, labourer
Arthur McDonnell, 30, labourer
Brian Burns, 35, labourer
Michael Murphy, 22, labourer
all except Mary and George born in Ireland.
In Glasgow Bridgeton in 1851 are
Peter Britten, 78
Agnes Duffey, 67
Mary Kennedy, widow, 34
James Kennedy, 9
Patrick Kennedy, 7
John Kennedy, 5
Mary Kennedy, 1
Bridget Farie, 77
Peter, Agnes and Bridget born in Ireland; Mary in Sanquhar, the children in Bothwell. I have seen Mary's death certificate, which confirms that Peter and Agnes are her parents.
By 1861 Mary has married her second husband, John Storrie, and they are in the parish of Shotts
John Storrie, 49
Mary Storrie, 45
Elizabeth Storrie, 20
John Storrie, 18
Helen Storrie, 14
Christina Storrie, 12
George Storrie, 5
James Kennedy, 20
Peter Kennedy, 17
Mary Kennedy, 12
Also in Shotts in 1861
Andrew Baxter, 56, stone quarrier
Sarah Baxter, 40
Andrew Baxter, 18, apprentice joiner, Shotts
James Baxter, 9
Can't find this Andrew jr in 1871. Could he be your missing one?
Now for a bit of a ramble which may or may not have anything to do with this query.
In 1841 there's a family at Gartsherrie in Old Monkland consisting of
Andrew Burns, 30, labourer
Mary Burns, 10
Margaret Burns, 8
Ann Burns, 6
Peter Burns, 4
Robert Burns, 3
Andrew Burns, 2
Katherine Burns, 20
Dommanick Burns, 20
John Galler, 18,
Felix Diver, 20
Thomas Campbell, 20
Dennis McGuire, 35
Anthony Gallaghan, 30
All the adults and Mary born in Ireland; all the adult males labourers
In 1851 there is a household in Coatbridge consisting of
Andrew Burns, 47, railway plate layer
Ann Burns, daughter, 17
Peter Burns, son, 15
Robert Burns, son, 13
Andrew Burns, son, 11
all born in Lanarkshire, except Andrew senior, who was born in Ireland
They might be the family of Andrew Burns and Anne Weir; their son Robert was baptised in St Andrew's RC, Glasgow in 1836 and their son Andrew was baptised there in 1838.
Andrew Burns, widower, 50, railway surfaceman, died in Sunnyside, Coatbridge on 23 August 1856. His father was Patrick Burns, blacksmith, and his mother was Mary Haggarty. The death was registered by Peter Burns, 15, son.
I haven't found either Peter or Andrew in the 1861 census. Nor can I find deaths of any male Burns with mother's maiden name Weir and born 1830 to 1840.