Bernard W Kane, 10, is with his father Bernard in the 1871 census, so not father Michael.

Your other option looks good: (1871) Bernard Kane, 6. b.Cheshire, Runcorn, stepson of John O'Day aged 26 lab., b.Ireland, who was married to Margaret aged 40 or 48 (crossed through - think 40) b.Ireland, and their daughter Mary aged 5 b. Runcorn, and 3 boarders, living at 21 Peel Street, Widnes.

I wonder whether 15 Penn St., Widnes with 4-yr-old Bridget was anywhere near Peel St.

If the right Bernard, whose dead father Michael was married to Margaret, her maiden name should be on those Runcorn birth certs.

Sue