Hull City Cartwright was initially a youth product of
Grimsby Town's academy before being sold to
Hull City joining at U13 level. He had a short loan with
Barton Town in 2019, with 10 appearances for the club. Returning to Hull City, he made his first senior appearance with the club in a 2–1
EFL Trophy loss to
Leicester City U21s in September 2020. He then joined
Gainsborough Trinity on a one-month loan in the
Northern Premier League, keeping four clean sheets in 6 undefeated outings with the club. He signed his first professional contract with Hull City on 2 February 2021. He made his professional debut with Hull City in a 1–1
EFL Championship tie with
Queens Park Rangers on 19 February 2022, coming on as a substitute in the 66th minute after a horror injury to the starting goalkeeper
Matt Ingram. On 25 May 2022, Cartwright signed a four-year contract with Hull City. On 17 June 2022, Cartwright agreed a season-long loan deal for the
2022–23 season with
Peterborough United under former Hull City manager
Grant McCann. On 16 January 2023, Cartwright's loan at
Peterborough United was cut short. On 26 January 2023, Cartwright joined
Wycombe Wanderers on loan until the end of the
2022–23 season. Cartwright would go on to make one appearance for Wycombe, a 3–0 defeat to
Cheltenham Town. On 4 July 2023, Cartwright returned to his hometown club, signing a season-long loan deal for
Grimsby Town. He stated upon his arrival "I'm really happy to be here. It's an area I know very well and I've been a massive fan of the club since I was a kid and this is something I've dreamed of from when I was a young fan. Cartwright was awarded the Young Player of the Year award during his season at Grimsby. On 7 August 2025, Cartwright signed for
Hartlepool United on a season-long loan deal. He made his debut for Hartlepool two days later, keeping a clean sheet in a 0–0 away draw at
Yeovil Town. He saved a penalty in the following match in a 2–0 home win against
Braintree Town. After an impressive start to the season, Cartwright was nominated for the National League Player of the Month award for August. Having been forced off with an injury on 1 October, his loan was terminated to allow him to return to his parent club for treatment. ==International career==