Some additional information from Australian Assisted Passenger records that might help…
John Hennesy age 27 occupation carpenter, listed with Elizabeth Hennesy age 21 occupation housemaid. Both Roman Catholic, John cannot read or write, Elizabeth can read & write, native place for both Co. Cork. The Gilmore seems to have left from London and appears to have provisioned on the way at Cape Town.
I dont think any records for this type of migration are held in Ireland, but there might be further details in England, e.g. National Archives.
Src : New South Wales, Australia; Persons on bounty ships arriving at Port Phillip (Agent's Immigrant Lists)
Added : I found a previous post on this couple : John Hennessy and Elizabeth Sloane / Diocese of Cloyne
(the marriage discussed in this earlier post is Church of Ireland...)