I had a busy day to day so wasn't here much. The rest of the children won't have birth certs as civil registration only started in 1864, but hopefully all their baptisms will be in St.Mary's Parish and it is just a matter of finding them there.
The marriage also won't have a civil cert, (assuming both parents are RC and the marriage took place in an RC church) the marriage is where you could come unstuck, marriages on that site for St.Mary's don't start until 1870, and they don't seem to be anywhere else
https://www.johngrenham.com/records/rc_church.php?churchid=0960&parish=St.%20Mary%27s,%20Droghedabut a marriage usually took place in the brides parish so if Ann is from a different parish it may be possible to find it.
Can you ask your cousin if the marriage year is assumed because of the year of the first child or was a record of it found.
Also the first child is born 1858 and the last 1866 which is less than 10 years, women often but by no means always had children over a 20ish year period so unless a marriage record was found there is always the possibility there are more children also the family could have started out some where else, keep in mind we don't know John and Ann's age when they married.