I would like to suggest that some parts of the Harris family may in fact been Mariners or of the Militia at this period. There are various churches in Cape Town in this period and some of the microform imagery appears on familysearch.org.
use the following link >
https://familysearch.org/learn/wiki/en/South_Africa,_Cape_Province,_Western_Cape_Archive_Records_(FamilySearch_Historical_Records)
or
google "familysearch.org South Africa" to see what sort of records are available to start looking.......
you may have to go to a LDS centre and hire some of those microfilms.
If Frederick was christened in the Cape then the image on the microfilm should give the parentage, and subsequent reading of the films other sibs may appear........ I would also consider land documents and probates from the Cape........if there were in fact any for Fredericks parents....
It should be noted that in 1797 the British sent 9 warships to the Cape; and in 1795 there was Battle of Muizenberg..........
Albert Denison Conyngham [https://en.wikipedia.org/wiki/Baron_Londesborough] should appear somewhere on the censuses [1841 & 1851] up until 1860 when he died, and William no doubt should also appear....... I am not certain of the coverage of York however so some kind soul may further that for you..... If those census entries are found then your Harris chap may appear in the household or nearby......
Once you can establish Fredericks age then certificate references such as marriage and death may be also found....... but, since the name is somewhat of a popular one you may have to eliminate a few possibles.......
There is however, a certificate reference .....>
Deaths for March quarter of 1852 Harris, Frederick Charles Henry registered York [9d 28]
as Lord Londesborough lived around York but I dont know how this is to be confirmed online without taking a risk and purchasing the said certificate..... I suppose the next option is to find a grave marker or headstone that may give some clues......
Others may have some ideas here.....
Furthering that the said Frederick was born about 1797, it would be, if indeed the event happened,
about 1820 or so that he could have married and possibly bore off-spring...... I would be looking for records possibly on familysearch.org suggesting marriages or christenings at this period ....... or if a census is found say 1841 a few if any children may still be at home........
I am not certain if any 1831 censuses were done around York or if done even survive in any form... so trade directories may give some indicators as an armourer would not be all that popular in comparison to say farm labourers..... so maybe that could be an option to exploring where the chap lived....
Furthering this , I would not limit Fredericks chances to being in York, and consider that he could have lived as far away as Greenwich or in Kent as a whole , because somewheres in my mind I recall a chatterer discussing their Harris genealogy in Kent......
I would also be looking for these folk on the National Archives website and whether any estates pertaining to the Lord survive today whereas a historical trust collection may be available or similiarly an archivist....... good luck with this