Hi Woodskid, I searched some subscription site and Irish Genealogy a free site for the birth and a possible marriage to no avail. Civil registration commecnced in Ireland in 1864 so the birth should be recorded. There are a number of Jeremiah Hayes births but no mother Elizabeth Hayes, there are various Dwyer / Hayes marriages but no Jeremiah birth to any of them. Can you pinpoint any other children to see if we could find the couple, mind you did they marry in Ireland. I ask as I am aware that the town you mention Peterborough was founded and a lot of Irish went there, I have had recent contact with two people from the area, Emily township who went in about 1830, there is a reunion planned for founding families there in either 2023 or 2025.
While it will not assist you in your search there is a site here that may interest you, https://tipperarystudies.ie/historical-society/
While both your surnames are common in Tipperary if you can think of any more information please revert and I will have another look. There are a few Tipperary Facebook pages if you do that genealogy but as you do not have an exact or close location it would be difficult to make a connection.
Hope something turns up for you.