I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse. By 1800, they are almost non-existant. Many B/M/D events either weren't recorded, or haven't survived.
For James Ward B 1806 +/- 5 yrs, Co. Meath, with a father forenamed John, mother Catherine, I found 1 record in 1806. However, the mother's surname is neither Farrelly, nor does it begin with "F."
There is also an 1806 record for the above with a mother forenamed "Judith," but the surname doesn't begin with "F."
A view (usually at a small fee) of the detailed record SHOULD show the mother's surname in both instances. Hope this helps. Good luck.