I didn’t find any information about Peter Starkey’s birth. But I thought you might be interested in the following…
When Peter and Catherine’s first child, Peter, was born on 22 Jul 1846, he was born in the poorhouse. This would have been during the Famine. See Microfilm of Parish Register: https://registers.nli.ie/registers/vtls000634201#page/19/mode/1up
Peter and Catherine Starkey had at least three other children as per rootsireland:
Starkey Patrick Francis 1851 Co. Galway (as you mentioned)
Starkey Mary 1854 Co. Galway
Starkey John 1857 Co. Galway
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In Griffith’s Valuation of Galway in 1856-57, there was only one head of household named Starkey:
Starkey Thomas T/galway/shop St. St. Nicholas Co. Galway
Source: rootsireland.ie.
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This might be your Catherine….
Name: Catherine Clogherty
Baptism: 23-May-1819
Address: Gortatleva
RC Parish of RAHOON, Co. Galway
Father: Patrick Clogherty
Mother: Bridget Flaherty
Sponsors: Alexander Brawan
Helen Brawan
Source: rootsireland.ie
I hope this is helpful.