There are only 4 O’Gara births registered in Ireland in 1922. None in Galway. One a male in Jan – Mar at Strokestown (Co Roscommon), others in Boyle (Bridget) , Castlereagh (Dominick) and Dunmanway (Peter).
Widening the search to the years 1915 – 1926, produced only a single half match – Columba Timothy Gara b Oct - Dec 1920 in Glenties (Co Donegal). (The O’ prefix is often detached in Ireland so it’s worth checking under Gara as well as O’Gara).
I also looked for William Patrick O’Gara (both for a marriage and his birth). I saw one Wm Patrick b Jan – Mar 1885 in Ballina, but who appears to have died in Ballina in 1890. Patrick O’Gara was more common and there were quite a few of those. No obvious marriage.
Searching for William Patrick O’Gara in the 1911 census, produced nil hits. There were 2 possible William O’Gara matches (of an age to father a child in 1922):
http://www.census.nationalarchives.ie/pages/1911/Dublin/South_Dock/Warrington_Place/86244/http://www.census.nationalarchives.ie/pages/1911/Sligo/Kilfre/Doon/752006/Patrick O’Gara is more common. Only one in Galway in 1911, but he was single and aged 47.
Not sure what to suggest next.