Seems a bit of a complicated family at first glance.
1871 Elizabeth Cunningham 45 widow with son James 8 listed as brother to head of household Thomas May, but is she actually his mother in Law and James his brother in law?
1881 Elizabeth Cunningham is with son James William Cunningham 17 and married daughter Elizabeth Doulton 32 and grandson Wilson May.
From GRO birth reg of James W Cunningham shows no maiden name, so he was probably illegitimate.