Hello,
I think I have found your family in the 1860 and 1870 census. (Did not find Henry in 1850.)
1860 New York City census Ward 22, Distr 1, Household # 2208 (children all born New York)
Henry Corlett 29 shoemaker England; Ellen 24 Ireland; James 5; Catherin 3; WIlliam H. 1/12; Rose Rogers 44 (?) Ireland; John Rogers 10 New York
(I would not be bothered by the fact that it says Henry is from England -- very common for census takers to write "England", even though what is given is "Isle of Man"
1870 New York City census Ward 22 Distr 20 (43rd st 10th & 11th ave) Household 554:
Jno Robb 44 laborer Ireland; Kate 36 Ireland; Geo 11 Ky; Martha 14 Ireland; Wm Barker 26 Ireland; Ellen Corlett 36 Ireland; Jas 16 NY; Kate 13 NY; Wm 11 NY; Chas 7 NY; Saml 3 NY
(Does this mean Henry had died by 1870, or was he living elsewhere?)
Good luck!
Lois