I search the IFHF's online database @ 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 nearly non-existant.
For Michael Wickham B 1834 +/- 5 yrs, somewhere in Ireland with the parents' names not used (a broad search), I found 0 records. This infers either this B event wasn't recorded, or the record hasn't survived.
I found nothing when I searched for Marg Navsy and Patrick Wickham. Ditto.
Suggest you visit the database, register (free), opt to be notified as additional records are added, and browse the database.