I propose to disconnect Joseph Burnham from the parents currently shown in Melton Mowbray, Leicestershire, and connect to John Burnham (1756-1834) of Alfreton, Derbyshire, (https://www.wikitree.com/wiki/Burnham-3676) as father.
The only records for Joseph in England are his marriage in South Normanton, Derbyshire, in 1820; the christenings of two of his children in Alfreton, Derbyshire; and his emigration in 1826 aged 28, pointing to a birth in 1798. There does not appear to be a corresponding birth record. The christening record identified as a “fact” on the profile is for a different person, born and died in Melton Mowbray in 1796 (https://www.wikitree.com/wiki/Burnham-3687).
The witness at Joseph's marriage in 1820 was a William Burnham. The same signature appears as groom at the marriage of William Burnham (https://www.wikitree.com/wiki/Burnham-3677) in South Normanton in 1815, and as witness at the marriages of John Burnham, Jonathan Burnham and Mary Burnham, all in South Normanton in 1811. John, Jonathan, Mary and William were all children of John Burnham (1756-1834). This suggests that Joseph is connected to this family.
There is a christening record for “John” son of John Burnham in Alfreton on 14 October 1798. I believe this is an error for “Joseph” son of John Burnham. I can find no subsequent record for a John born in 1798; and there was already an older brother John born in 1784 (the one who married in 1811 – https://www.wikitree.com/wiki/Burnham-3678).
Am I missing anything?