Barnshaw began his career with hometown club Hebburn Argyle, and remained with the North Eastern League club until 1913 when he signed for The Wednesday. Injury prevented him breaking through at the First Division club and he returned to Argyle, though he was soon back in Sheffield, signing for United for a fee of £90.

He made three Division One appearances for the Blades but failed to make the grade and signed for Watford in 1914 in a £50 deal. Barnshaw played eleven times as the club won the Southern League First Division title in 1915.

Leeds City invited Barnshaw to appear as a wartime guest for the club on Christmas Day 1916. Regular left-half Levi Thorpe was injured and Barnshaw deputised when Leeds beat Bradford City 1-0 at Elland Road.

He appeared again a week later when City lost a friendly to Liverpool, but suffered an injury and never appeared again for Leeds, guesting instead for Darlington.

After the war, Barnshaw returned to Watford for a while before joining Third Division South club Aberdare Athletic in 1921. After one season in Wales, Barnshaw retired from playing.

Barnshaw died on 30 January 1974 in Watford.