Donald
Census records before 1901 are not available in Ireland so varius substitutes have been used.
CHURCH BAPTISMS are the most reliable as the local parish priest would know most of his parishioners so records there are reasonably accurate. Some larger towns had records going back to the late 1700's but most parish records begin after 1828 when Catholic Emancipation was achieved which allowed catholics to build churches.Unfortunately your search pre dates 1828 but you may be lucky if your parish records commenced earlier..ROOTS IRELAND has a good database of baptisms.
I failed to find any Clohessys born to a mother Anna,Anne,Johanna etc The only other record that may give a clue is the Tithe Applotment Records of 1830 but the more info would be needd to search that.Have you any record of Johns father,grandfather ,other relatives in Limerick etc. Was Anne a widow or did her husband emigrate ahead to find accommodation etc
The clohessey name is common in county Limerick.
The fact that I could find no common father to the four children I suggests they were baptised in a parish church where records began after 1828..That no children had Anne recorded as their mother also suggests a later parish..If you can provide more info I will look further
Regards
Donie O Sullivan