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After a number of searches on the subscription site Roots Ireland, I think your Mary Downey was baptized and married in a parish that does not have baptismal records back to 1820 and marriage records back to 1841. There are at least twenty parishes in this category. Also did not find an earlier baptismal record for a James Hanny.
I looked at the 1855 Griffiths Valuation head of household listing just to get a sense of where the surname Hanny was found since it is not a common surname like Downey. There were only eight records: three in Killimorbologue parish, two in Killeenadeema parish, two in Kilmalinoge parish and the final record in Lickmolassy parish. All of these parishes are in southeast Co. Galway and three border each other. The RC parishes that correspond to the civil parishes all have baptismal records starting in the 1830s. There is one RC parish in the area (Ballinakill) that has late marriage records (1854) so my conjecture is that Mary Downey came from Ballinakill civil parish. She likely lived not far from James Hanny who may have been in Killeendeema civil parish. Almost all marriages were held in the bride's RC parish
This is only a lead based on the records available. You need to be aware that there are actually four Ballinakill civil parishes in Co. Galway. One in the northwest, two in the northeast part of Galway and finally the one I mentioned in southeast Galway.
Let me know if you have any questions. I know my explanation is confusing and it might help if you looked at this map from johngrenham.com https://johngrenham.com/browse/county_civil.php?county=Galway Keep in mind that RC parishes boundaries differ from the civil parishes.