The child’s mother would be John’s first cousin.
The father was John's first cousin on their paternal sides (known)
The mother was not a first cousin to John (known).
The mother could possibly be a cousin to John's wife, but as I already said, that would be a coincidence given the known Roche relationship.
But I will now investigate a Hayes possibility for Sarah, in Dublin and Wexford.
All very strange. Strange in several successive generations.