I did just what you did.

Given the name Mary Barry and her four children - problem find her husband and maiden name.

Best place to start is to try and find birth or baptism of the children.

Albert is not a very common name so that is where I started. Bingo - name of father and mother.

We seem to be able to trace William Barry through 3 censuses and a burial.
He appears to have two wives, Jane & Mary.
Couldn't find marriage records for either and no burial for Jane.
Given all three are of Irish extraction these records could be in Ireland.

I have come across several instances of persons with two forenames using one as a child and then using the second later in life, usually after a major change in their life, such as marriage.

Hence the suggestion that Mary and Jane may be the same person.

All this does is to suggest possible lines for further research.