The name Hodgson does not appear on William's Birth, Marriage or Death Records and the middle letter H only appears on one census.
According to the 1851 census Williams grandfather Joseph was born about 1775 Bromfield, Cumberland
Joseph Barnes, Parents Daniel & Isabella
Baptism 7 Aug 1774 Bromfield, Cumberland
Rebecca was born about 1782 Great Salkeld, Cumberland, I cannot find a baptism or their marriage, however some of their children’s baptisms name her as Rebecca Idle. Williams mother below.
Ann Barnes Christening 30 May 1808 Bromfield, Cumberland Parents Joseph Barnes, Rebecca Idle
Daniel Barnes
Marriage Date: 11 Nov 1773
Marriage Place: Bromfield, Cumberland
Spouse: Isabella Raper
Colin