Hi,
There is a James Finnigan born 1841 dying aged 68 in 1909 in Leicester Volume 7a page 131.
Also, a marriage of James Finnigan in 1873 in the 4th qtr marrying either Mary Ann Thompson or Margaret Melia in Leicester Volume 7a Page 550.
Census 1881
James Finigan born 1843 aged 38 born in Ireland working as an Agricultural labourer living at 25 Abbey Street Leicester St Market. Living with wife Margaret aged 38 born in Whitwick and son Michael aged 7 daughter Bridgett aged 5, daughter Mary aged 0 (all kids born Leicester) and Patrick brother aged 42 working as a General Labourer born in Ireland.
Census 1891
James Finnegan widower aged 48 Coal Heaver born in Galway living in Baker Street Leicester with son Michal 17 Shoe finisher, son John 7 scholar, daughter Bridget 15 Hosiery hand and daughter Mary 10 scholar.
Census 1901
James Finnigan living at Benford Street Leicester, aged 53, widower, born in Ireland in 1848, farm labourer. Living there with son John (17 farmers waggoner), daughter Mary A (20 hosiery brusher) and a Boarder Sarah Cahill (17 hosiery seamer).
Patrick Finnigan born in 1840 dying aged 47 in 1887 in Leicester Volume 7a Page 134.
I also noticed in the Leicester Journal dated 17 September 1841 it mentions another brother Michael.
Hope that helps
Regards