Hello Spud Murphy, well this is a task and a half really; the details are very limited but I did manage to find five baptismal records for Michael Murphy in the Cork area:-
Church Baptism MurphyMichael Co. Cork 1816
Church Baptism Murphy Michael Co. Cork 1824
Church Baptism MurphyMichael Co. Cork 1824
Church Baptism Murphy Michael Co. Cork 1824
Church Baptism Murphy Michael Co. Cork 1822
Each of these Michael's had a mother named Margaret but there is NO records of any of these ladies having been registered in Cork within a five year period of 1781. No Michael born 1821 but if it is the census that you are working from, it is natural to have the incorrect year. The name Murphy brings up 1000's however and we really would need to have more details such as the father's name or a hint perhaps: is the 1781 date correct or just guesswork? I would so love to help you but even one more piece of evidence to give me alead would be great.
You can contact me in reply to this message killeedy1@irelandxo.com
Stay well. and Good Luck in your research. Jennifer Jean