It's been a long time. I still can't connect Patrick Alfred Durkin directly to Bridget Durcan O'Malley. But I did find some poosibilities for tracing Bridget back perhaps.
I started looking for young James O'Malley on the 1901 census he had been living with "uncle" Patrick Durkin in Liverpool and was 8.
So this is what I found. Turns out to no surprise he became a butcher. But now he was with another Uncle ...Michael. What did surprise me was that Annie & Patrick were there. They are on the Irish census in Mayo with Bridget for 1911. But here they are again.
@ Address: 4 Edinburgh Rd Registration & Civil District: West Derby Civil Parish West Derby
Forename Age Gender Relation Marriage Status Years Married Occupation
Durkin Michael 36 Male Head Widowed 2 Butcher
Durkin Martin 10 Male Son Single 0 School
Durkin Margaret 30 Female Sister Single 0 Housekeeper
Durkin Ellen 26 Female Sister Single 0 Shop Assist
Omalley Annie 24 Female Niece Single 0 Shop Assist
Omalley Patrick 23 Male Nephew Single 0 Pork Butcher
Omalley James 18 Male Nephew Single 0 Pork Butcher
Looks like his wife died after only 2 years of marriage. I did find reference to a Michael Durkin marrying a Margaret Walsh Apr-June 1901
So I looked back to 1901 census for Michael & I think this is him. Earlier you had helped me to find a marriage which my grandmother Annie O'Malley was a witness to.
Helen (or Ellen) Durkin and Thomas Peter Walsh 25 Oct 1913 Sacred Heart
witnesses: Ann(a) O'Malley 78 Soho Street , Frederick Murphy Gt Howard St
The head of house where Michael lived was Michael Walsh and his son was Thomas Walsh. Possibly this Thomas Peter Walsh..don't know
So I had found 4 Bridget Durcans born between 1859 & 1864. This is the only one related to the Walsh family. Perhaps Uncle Michael Durkin may be Bridget’s brother. But I don't know.
Her marriage record to William O'Malley 22 Feb 1883 said she was 24 so she was born approx 1859-1860. But the 1901 &1911 Census records and her death record indicate she was born around 1864. Later records are sometimes less true. Perhaps she told her second younger husband she was younger than she actually was and continued to stick to it.

?? Is this my Bridget- Still no relation shown to Patrick Alfred Durkin
Bridget Durkin b. 14 June 1860 Christened 30 June 1860
Father Martin Durkin Mother: Margaret Walsh
Sponsors: Cate Walsh/ Michael Durkin
Address Riverstown Parish Kilmoremy/ Ballina
Siblings:
Mary 1856 (Bridget named a daughter Mary)
Ellen 1858 (Bridget named a daughter from second marriage Ellen)
Cate 1862 (Bridget named a daughter Catherine)
Mary 1856 (Bridget named a daughter Mary)
Martin 1866
Sarah 1868
Margaret 1870
Dominick 1872
Michael 1873
Anne 1875 (Bridget named a daughter Anne)
Kind Regards,
Anne