Going by the family tree I found on ancestry, the first two children were
James 1869, and
Elizabeth ?
If these where named for the paternal grandparents, then a candidate set of parents for James can be identified, from births of probable siblings of his in Gorey, viz:
James Kavanagh, and
Elizabeth Sheehan
They had multiple children in Gorey parish, as follows
Michael 1845
Elizabeth 1848
Margaret 1851
Anne 1854
There may be more - I only searched 1845-1855.
But note Michael 1845.
Address is Gorey - not sure how that fits with father James being a farmer as per 1868 marriage rgistration.