The 1st question would be what denomination Protestant or Roman Catholic.
2nd would be where in Fermanagh.
If Catholic then there are no records before 1835 for Fermanagh (with the exception of Drumlane 1810).
If Presbyterian most written records don't go back that far due to the Penal Laws.
If Church of Ireland (Anglican) records may or may not exist and may or may not be online (some were in Dublin Record Office and lost 1922
There are some local sites eg
https://www.igp-web.com/IGPArchives/ire/fermanagh/church.htmhttps://www.cotyroneireland.com/ or
https://cotyrone.com/cofermanagh/index.htmlFor all other surviving Church of Ireland Records you are talking microfilm in PRONI, scrolling through and reading.
Most military service records, and they are not many for the early 1800's, were deposited in the National Archives, Kew, London and are on either Ancestry or Findmypast, there may be pay rosters etc stored there but they would have no family detail. Even the service records that do exist around then do not list any next of kin just an birth parish, attestation place and discharge place.
The thing is he was not the only Hugh McCaffrey eg one was born Enniskillen & served 1st Bn 88th Regiment of Foot 1809 to 1814 discharged with amputated arm aged 24. A Hugh McCaffrey of 27th Foot was a POW in Givet, France 1812. Another Hugh McCaffrey of Clougher, Tyrone was discharged with wound from 3rd Bn 1st or (Royal Scots) Regiment of Foot 1816. There would have been other's farm labourers, shoe makers etc, from baptisms you have no way of distinguishing.
Only a select few wifes accompanied the regiment, they tended not to pay for that or provide accomodation.
https://blog.forceswarrecords.com/service-in-the-british-army/There were 2 Hugh McCaffreys in the 27th foot both awarded Army Gold Medal / Military General Service Medal, 1793-1814 one with 3 clasps and one with 6 & on the Colonial List. Peninsular Medal Roll 1793-1814 Findmypast.
There is a service record for a Hugh McCaffrey in 27th Foot, but decade younger, enlisted 1808 aged 16 but served 24 years discharging with pension 1831 back to Enniskillen. For a service record to survive he would have to have been pensioned on completion of service c.21 years/discharged wounded.