Patricia:
I searched the subscription site Roots Ireland and was unable to find any McDonough baptismal records in Co.Galway where the father was John and the mother had a surname of Molloy. Based on your note, I assume Patrick was born around 1850. There are a good number of RC parishes in Co. Galway where records start later than 1850.
Did Kate also emigrate?
Have you tested your DNA?
Roger McDonnell