I'm not sure how much of this info you already have, but Ballyhaunis is in the Catholic parish of Annagh. The parish records which are available online go back to 1851 for baptisms, so you should be able to find some or perhaps all of the children there. The records are available at this link:
The marriage records go back to 1852 (with a few earlier ones), so if the birth year you have for Mary Lyons is correct, you may be able to find her marriage record as well. In case youu need to look for anyone in a nearby parish, you can use the interactive map at that site to move to a different parish. Claremorris (Kilcolman) and Bekan have baptismal and marriage records going further back.
If you find a relevant baptsmal record, it will usually state the name of the townland where the family lived. You can then locate more info for that townland at this site about Ballyhaunis, which lists all of the townlands and has links to more info about them: