This Robert is possibly the first cousin of another Robert Gardhouse, who was born one year later.
According to a secondary source, the William 'Guardhouse' listed in the 1851 census was a son of Robert and Ann's, and he married Catherine Meally (1822-1904).
Sources
"England Births and Christenings, 1538-1975," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:JMNH-YR4 : 11 February 2018, Robert Gardhouse, 28 Sep 1789); citing LANERCOST,CUMBERLAND,ENGLAND, index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 564,347.
"England Marriages, 1538–1973 ," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:N2KY-WTT : 10 February 2018), Robert Gardhouse and Ann Holmes, 02 Jan 1813; citing Lanercost,Cumberland,England, reference , index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 547,190, 564,347, 90,641.
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DNA Connections
It may be possible to confirm family relationships with Robert by comparing test results with other carriers of his Y-chromosome or his mother's 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 Robert: