Hi Andrea
If you search for William Wrights who served in the RA on the NA Catalogue -
http://www.nationalarchives.gov.uk/catalogue/search.asp - you get 11 hits. You will be able to discount half from the dates they served. The problem is these probably only represent about a third of those of that name that served. The rest will have been lost.
Born BRAMFIELD, Suffolk Served in RA Discharged aged 35 1826-1841
Born BALLYRONIE, Down Served in RA Discharged aged 38 1833-1854
Born RIVER, Kent Served in RA Discharged aged 40 1832-1853
Born ABBEY PAISLEY, Renfrewshire Served in RA Discharged aged 41 1824-1845
Born HACKNEY, London and Middlesex Served in Royal Horse Artillery Discharged aged 37 1799-1818
Born BRADFIELD, Yorkshire Served in RA Discharged aged 46 1808-1835
Born BALLYMORE, Armagh Served in RA Discharged aged 43 1812-1833
Born CHADLEY CORBET, Worcestershire Served in Royal Horse Artillery Discharged aged 45 1811-1834
Born WEDNERPOOL, Nottinghamshire Served in RA Discharged aged 43 1815-1839
Born WOOLWICH, Kent Served in 84th Foot Regiment; RA Discharged aged 43 1814-1834
Born DERBY, Derbyshire Served in Royal Horse Artillery Discharged aged 30 1805-1816
The dates are when they served.
The army simply did not record FH information at this time, so you will probably not find Lucinda mentioned anywhere. If you order these records – at 40p a page, say £10 for half of the above - you might find a man who served in Newfoundland at the right time. These papers should give his parish of birth. Failing that they all appear to be on the same microfilm – WO97/1271 – so you may be able to borrow a copy of the film under the A2A scheme and have it sent to your nearest LDS centre or other FH research facility.
Another course would be to visit Kew and look up the Muster Books (WO12). It may take you half a day to find the right battalion (there were nine plus a horse brigade), and a further half day to track his service. There is a small chance his first and/or last entry may give his parish of birth, or you can see where the sergeants were out recruiting when he enlisted. The muster will tell you where William was on the first of each month, so by tracking where his was before 1822 you can search for their marriage in nearby churches.
Lucinda should be easier to research. Her death certificate for example may give you some information about William.
Ken