These are the only Polkinghornes I can find in 1841 (in 1851, the Head is a William, Seaman, Pob Not Known)
Sadly, no sign of a Matthew in either year ..
HO107/413/1; Portsea Town; District: 1; Folio: 15; Page: 24
Kingston & Landport - 'Little Charlottes Row'
All born in County
Elizth Pelkinghom [sic] 30 'Wife of a Sailor'
Harriott Pelkinghom 15
William Pelkinghom 10
Frances Relkinghom 5 dau?
Sarah Pelkinghom 4
Mary Pelkinghom 2