Originally from Grimsby, Cartwright joined Hull City at Under-13s level before going on to rise up through the academy.

Having been a regular at various youth levels, the highly-rated goalkeeper was handed his senior debut by Grant McCann in the EFL Trophy against Leicester City Under-21s in 2020/21.

He signed a new three-year contract in February 2021 and enjoyed a season of progression in 2021/22.

He made his City league debut in the Championship draw at Queens Park Rangers in February and started the home game against Barnsley three days later.

Earning his first England Under-20s cap in March, Cartwright kept four clean sheets in six unbeaten matches during a month-long loan stay at non-league Gainsborough Trinity.

He was rewarded with a new four-year contract in May 2022 and reunited with McCann at Peterborough United a month later, joining Posh on a season-long loan.

Injury problems restricted him to one appearance in the EFL Trophy for Posh before returning to City in January 2023. Later in the month, Cartwright joined League One side Wycombe Wanderers on loan for the remainder of the 2022/23 season, making his League One debut against Cheltenham Town in April.

In July 2023, Cartwright returned to his hometown club Grimsby Town on a season-long loan deal. Keeping six clean sheets in 32 appearances, the shot-stopper won the Mariners' three Young Player of the Year awards in their end of season awards.