Welcome to Ireland Reaching Out!
The Clifden RC baptismal records start in 1838 so if your family was from that parish, baptismal records should have appeared on the subscription site Roots Ireland. However, I got no record when I made my search. Unfortunately, it appears your family was from Moyrus civil parish which means Roundstone RC parish where baptismal records start in 1872.
The 1855 Griffiths Valuation head of household listing for Moyrus civil parish shows three Michael Gorham records https://www.failteromhat.com/griffiths/galway/moyrus.htm One of the three is likely your ancestor.
Did any of the brothers stay in Ireland?
You may want to add Mary and her brothers to our XO Chronicles site https://irelandxo.com/ireland-xo/history-and-genealogy/ancestor-database