I'm not positive, but I think that Richard was probably the illegitimate child of Mary Armitage. In the 1851 census, he was listed as the grandchild of Martha Armitage, and Mary was the only one of Martha's children old enough to be his mother. At first I thought that it was a mistake, and he was actually Martha's son, but then he's listed as Frances' nephew in the 1861 census.
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Sources
1841 England Census, Lockwood, Huddersfield, Yorkshire, England. (Ancestry subscription required)
1851 England Census, Lockwood, Huddersfield, Yorkshire, England. (Ancestry subscription required)
"England and Wales Census, 1861," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M7C7-31S : 13 December 2017), Richard Armitage in household of William Broughton, Huddersfield, Yorkshire (West Riding), England; from "1861 England, Scotland and Wales census," database and images, findmypast (http://www.findmypast.com : n.d.); citing PRO RG 9, The National Archives, Kew, Surrey.
1861 England Census, Huddersfield, Yorkshire, England. (Ancestry subscription required)
This family in the 1881 census might be the same Richard Armitage:
"England and Wales Census, 1881," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:Q27R-LLJZ : 11 December 2017), Richard Armitage, Lockwood, Yorkshire,Yorkshire West Riding, England; from "1881 England, Scotland and Wales Census," database and images, findmypast (http://www.findmypast.com : n.d.); citing p. 3, Piece/Folio 4387/74, The National Archives, Kew, Surrey; FHL microfilm 101,775,229.
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DNA Connections
It may be possible to confirm family relationships with Richard by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Richard: