There is no known connection between the so-called John Bertie Cotton (actually John Cotton of Bertie County) and the Cotton family of Connington in England. Although I haven't got the details to hand, it is well known that some of his ancestry was an invention.
Sir John Cotton of Connington (1621-1702), grandson of Sir Robert Cotton (sometimes called Sir Robert Bruce Cotton) DID NOT marry Anne Hutchinson.
He married twice:
1. Dorothy Anderton;
2. Elizabeth Honywood. No issue recorded.
By Dorothy Sir John is recorded as having had a son, John Cotton, Esq. of Connington (died 1731-1, age 52). He married Elizabeth Herbert and they had two daughters, Anne (1677-1714) and Catharine )died an infant).
I hope this helps a bit.