Co. Clare's genealogy office does not belong on the History Foundation for online birth and marriage records.
You might be able to find them in the Tithe records which date from 1820-30's, these are located at the National Archives in Dublin but also now on line with Ancestry. Also located at the Clare library.www.clareroots.ie
a Heritage and genealogy resource
Griffith Valuations www.askaboutireland.ie
This is an example from 1852 too late for your Cornelius
but this one still in Co. Clare www.askaboutireland.ie
Eg: Cornelius Murphy, Parish of Kilbkennedy, Townland of
Kilmore Union of Limerick, tenant of Timothy O'Brien
7a house outbuilding and land 8 acres 1 rood 12 perches
There are 333 other Murphy's in Clare.
Ennis is a Poor Law Union Town, and is also 'Ennis Part of' is a Townland. More than 6 parish's for the Union but these 3 seem the more likely: Parish Clareabby, Parish Drumcliff and Tomfinlough.
There are still 3 Cashman's in the telephone book and about 600 Murphy's, addresses given are sufficient for mail delivery.