I can't see some of the information you mentioned in the 1881 census such as Patrick's mother or the first child Sarah 1860.

If you analyse Brian Lawsons BMD it will add information

www.lawsons.ca

The marriage section shows the marriage of Patrick and some of the children, the birth section shows an extra child Margaret 1881 and children of children.
The death section has Patrick as Mcconery.

There is still a Mccovey in the telephone book who may add something.

Len Faragher