Wright was born at Quarndon, Derbyshire the third son of Ichabod Charles Wright of Watnall Hall, Nottinghamshire. He was educated at Brighton College and admitted to Trinity College, Cambridge on 29 January 1858. He was a scholar in 1861 and also that year rowed in the winning First Trinity Boat Club coxed four which won the Stewards' Challenge Cup at Henley Royal Regatta. He was admitted at the Inner Temple on 20 August 1862 and called to the bar on 30 April 1866.
Wright was a member of his father's banking firm in Nottingham from 1867 to 1878. With his father he published a selection of psalms in verse. He translated the Iliad, I-IV into English hexameters, and the Aeneid, I-VI into blank verse.
Wright stood unsuccessfully for Nottingham South in 1885  but was elected Member of Parliament (MP) for South Nottingham at the 1886 general election. He held the seat until he stood down in 1895.
Wright married Mary Jane Cartledge, only daughter of William Cartledge, of Woodthorpe in 1865. He married secondly in 1869, Josephine Henrietta Wright, his first cousin, daughter of the Rev. John Adolphus Wright, rector of Ickham, Kent, and had four sons and one daughter.
- "Wright, Henry Smith (WRT858HS)". A Cambridge Alumni Database. University of Cambridge.
- R C Lehmann The Complete Oarsman
- Good Reads Henry Smith Wright
- Homer H S Wright The Iliad I=IV
- Debretts House of Commons and Judicial Bench 1886
- Craig, F. W. S. (1989) . British parliamentary election results 1885–1918 (2nd ed.). Chichester: Parliamentary Research Services. p. 163. ISBN 0-900178-27-2.
- Nottinghamshire History and Archaeology
- Hansard 1803–2005: contributions in Parliament by Henry Smith Wright