The date 1844 for son Patrick is pretty early for Galway.
Galway is divided genealogically into east and west but both belong to an Irish History Foundation with birth and marriage records. Marriage records
have only both of the couple's father's name, place/parish
(usually bride's parish)and a date.
With both names of the couple you can search for the children as a group.
Women's disappear as they marry and take a husband's name.
So that leaves John Doyle. Doyle as a surname is difficult as there are usually just too many people with that surname.
Hussion is an unusual name so you should have no problem finding birth records if the records in the parish has
survived. Certainly you could find the marriage record.
There are 0 Hussion in the current on line Galway telephone book, but 105 Doyle's.
In 1856 the Griffith Valuation took place in Galway and
Patrick and Bridget may have been occupiers of land.
The name Hussion is listed 23 times in the Valuation in Galway and all 23 are in Galway. So definitely Bridget was from Galway. There are 4 Hussion's in the Parish of Kilconnell. Two first names of Patrick or William.
There is 1 Hussion in an 1846 Dublin Directory named Andrew
of Prospect Hill, owns a Public House.
There were 48 Doyle's in Galway in 1856 and only one in the Parish of Kilconnell, Townland Barnavihall, PLU Ballinsloe
Bridget Doyle, Parish of Kilconnell, Townland Barnavihall
tenant of Rt. Hon. R.W. Greene
#1 herd's house, outbuildings and land
For Bridget to have a land lease in her name she is probably a widow, this fits your information.
Although the above doesn't answer the question of Bridget's husband's first name, the fact she has a son Patrick. If he is the first child he would either be named after his father or grandfather. The 2nd. son would be named after her father.