I don’t recognise Leaghretty, either as a townland or a parish in Armagh. It could be Loughgilly corrupted. Worth looking at.
If Samuel Templeton was a farmer and alive in the 1830s he should be listed in the tithe applotment records. Most people with land were listed there. There were some exceptions notably people farming on church land who were often exempt but most would be listed. There were only 3 Templeton farms in Co. Armagh in 1828-1834, in just 3 parishes. List below:
Templeton, George-- Townland : Farragh Year : 1828-Drumcree-Armagh
Templeton, Thomas-- Townland: Ballainen Year: 1834-Loughgilly-Armagh
Templeton, Widow - Townland: Ballainen Year: 1834--Loughgilly-Armagh
Templeton, William-- Townland : Terryscane Year:1833-Eglish-Armagh
No Samuel but there was a widow T in Ballainen (more often spelled Ballenan). Her husband might have been a Samuel. You might want to check the parish records for the above parishes to see if you can trace your family. You need to know their denomination. I looked at the 1901 census for Armagh. There were only a handful of Templetons in the county. All were Church of Ireland save for 1 Brethren family (though they wouldn’t have been Brethren in the 1700s. The denomination was only established in the 1820s and didn’t really get going till the 1859 religious revival).
I’d focus on Church of Ireland and possibly Presbyterian records. Loughgilly Church of Ireland has records starting in 1804. There is a 2nd church in the parish at Belleek but it’s early records were lost in the 1922 fire in Dublin. Mountnorris Presbyterian has baptisms from 1810 and marriages from 1804. Tullyallen Presbyterian has records from 1792.
Drumcree Church of Ireland has records from 1788. There is also a 2nd COI church in Portadown but its records don’t start till 1826. The Presbyterian church does not have any records before 1868.
Eglish Church of Ireland has records from 1803. The local Presbyterian church in Knappagh has no records before 1900.
There are copies of all the above records in PRONI (the public record office) in Belfast, save for Tullyallen Presbyterian where the only copy of the records is still held by the Minister. So you would need to write to him/her if you want to search them.
Henry is a more common surname in Armagh. A mix of RC, Church of Ireland, Baptist, Brethren, Presbyterian and Reformed Presbyterian. So take your pick.
There was only 1 McCandlish in Co Armagh in 1901 and she was born in Co. Down (she was Presbyterian). So that might make you wonder whether John Templeton’s marriage to Agnes McCandlish took place outside Co Armagh. Belfast perhaps.
Statutory birth registration didn’t start in Ireland till 1864. Prior to that you have to rely on church records. To do that you need to know the denomination and location and even then not all churches have records nor are they all on-line. It can be a bit of a needle in a haystack.