Civil registration of births started in 1864 so we would have to rely on available church records.
I searched on the subscription site Roots Ireland and there were no 1850 baptismal records but four records in the 1840s with father Thomas. The info is shown below and since you don't know the mother's name, we can't confirm any of these records but at least you have some leads.
Thomas Cary baptized February 10 1843 Rathcormac RC parish father Thomas Cary mother Catherine Connell
Thomas Carey baptized May 8 1845 Youghal RC parish father Thomas Carey mother Ellen Cashman
Thomas Carey baptized December 28 1845 Kilwoth RC parish father Thomas Carey mother Honora Burke
Thomas Carey baptized June 16 1845 Killavullen RC parish father Thomas Carey mother Margaret Ryan
Are you aware of any siblings?
As you look at your matches and if they have a tree, I would look for the four maiden names above for a possible connection.
Let me know if you have any questions.