Hi Cluin McAlear:
There is a John Hennessy baptised in Miltown Malbay parish in April 1832 to Owen Hennessy and Bess Mulqueen from Caherogan townland. Baptism sponsors were not noted, but the image for this record is available at the NLI parish registers website here: https://registers.nli.ie/registers/vtls000634725#page/9/mode/1up It is on page 9 at the right hand side of the page.
I did not find any other reference to children born to this family in the Miltown Malbay transcribed registers which are available to view on the Clare Library website: http://www.clarelibrary.ie/eolas/coclare/places/kilfarboy.htm You will find a lot of useful information to go through on this link including parish registers, Griffith's Valuation records and Census information.
You may be able to trace Owen and Bess Hennessy forward in Griffith's Valuation and then through the Civil records which are available online. If they died after 1864, their deaths may have been registered. The link for that site (which is free) is here: https://civilrecords.irishgenealogy.ie/churchrecords/civil-search.jsp
You may also be able to find more information about this family through the Graveyard transcriptions for the parish which have been done (possibly not complete). They are located at the link in the 2nd paragraph.
If you need further assistance, please let me know.
All the best,