Kitching began his career at Leeds United and signed a two-year contract with the club in July 2017.[6] After featuring regularly for Leeds' under-23 side during the first half of the season, Kitching joined Harrogate Town on a one-month loan in January 2018.[7][8] His first appearance for Harrogate came as a substitute in a 4–1 victory over Stockport County before making his first start for Harrogate on 10 February 2018 in a 3–1 victory over Leamington, in which he scored the first goal of his senior career.[3][9] On 16 February 2018, his loan was extended until the end of the season.[10] After making 9 league appearances for Harrogate, scoring three goals,[3] Harrogate were promoted to the National League after defeating Brackley Town 3–0 in the play-off final.[11] On 20 July 2018, Kitching signed a new contract with Leeds United before rejoining Harrogate on a six-month loan deal.[12] In January 2019, his loan was extended until the end of the season, after scoring three goals in 25 appearances during the first half of the season.[13] Across the 2018–19 season, Kitching appeared in 34 league games for Harrogate, scoring twice.[3]
Forest Green Rovers
Kitching signed with Forest Green Rovers in July 2019 on a three-year contract for an undisclosed fee.[14][15] He made his debut for Forest Green Rovers as a substitute in their 1–1 League Two draw away to Walsall,[16] before making his first start on 13 August 2020 in a penalty shoot-out victory over Charlton Athletic following a 0–0 draw in the EFL Cup, in which Kitching conceded a missed penalty in normal time.[17] He appeared in 29 of Forest Green Rovers' 36 league matches across the truncated 2019–20 season.[3]
Barnsley
On 5 January 2021, Kitching signed for Championship club Barnsley on a four-year deal for an undisclosed fee believed to be in the region of £600,000.[18][19][20] He scored his first goal for the club on 13 September 2022 in a 1–1 draw against Port Vale.[21]
Coventry City
On 1 September 2023, Kitching transferred to Championship side Coventry City.[22] On 14 February 2024, he scored his first goal for Coventry, a stoppage-time equaliser in a 2–2 draw against Plymouth Argyle.[23] On 1 April, he scored two own goals in a 2–1 home loss to Cardiff City.[24]