Have you looked for Patrick in Irish birth records at irishgenealogy.ie? It's a free website.
Have you looked for him at rootsireland.ie, which has birth and baptismal records, among others? There's a fee to use it.
Also, have you checked all Patrick's records in Merseyside? Census, marriage (church and civil), death, headstone, obituary/death notice, probate, etc.?
You might also consider joining an Irish genealogy society such as the Irish Family History Society or the Irish Family History Forum.
Finally, don't put too much stock in the birth year being 1870. People were not so aware of birthdays as we are, and you'll sometimes find a variety of ages for the same person in the records.