Born in Chesterfield, Jessop started his youth career at Sheffield Wednesday before signing for Scunthorpe United as an under-10 in December 2011 following his release by Wednesday. After a two-year scholarship with the club, he was offered his first professional contract in summer 2020.
Jessop joined Northern Premier League Division One North/West side Pickering Town on a one-month loan in October 2020. He was recalled shortly before the end of his loan spell due to the suspension of Northern Premier League fixtures as a result of a second national lockdown in England, having scored 3 goals in 7 starts for the club and was named their Player of the Month for October. He made his senior debut for Scunthorpe United on 13 April 2021 as a 71st-minute substitute. He made 3 appearances for Scunthorpe in his debut season in professional football
After three appearances for Scunthorpe in August 2021, it was announced that Jessop had joined Gainsborough Trinity on a month-long loan on 10 September 2021. He made his debut for the club on 11 September 2021 as a substitute in a 1–0 defeat to Warrington Town.
On 8 March 2022, Jessop joined National League North side Farsley Celtic on loan for the remainder of the 2022–2023 season.