Clitheroe born midfielder Clarkson signed his first professional contract with Liverpool on 22 July 2019 and was on the bench for the first team two months later in the league cup. Matt Addison, Liverpool Echo's expert on the club's youth teams says about the promising youngster: "Clarkson kind of plays as a number six – but more of a Jordan Henderson number six than a Fabinho. He’s very good on the ball, and he has a brilliant eye for a pass." Clarkson joined Liverpool’s Academy at the age of six in 2008. He came through the youth ranks at Kirkby, progressing all the way to the under-18 and under-23 sides before making his senior debut for Klopp’s first team on 17 December 2019 against Aston Villa in the League Cup.

On 9 December 2020 Clarkson played all ninety minutes of Liverpool's Champions League group game against Midtjylland in Denmark. He was also on the bench for three Premier League games during the 2020/21 season but did not get on the pitch. On 16 August 2021 a season long loan with Blackburn was agreed. Clarkson expressed delight at the opportunity, telling their website; "This was the only club where I really wanted to come when I knew I had the chance to go out on loan. I grew up not far from here, I used to go to the games from when I was two until I was about 16 and I also played here when I was younger, so I've always supported the club and it feels good to be here. I've got a chance this season to prove myself and to be able to do that in front of the Rovers fans, at this club, is the best outcome I could have had." It didn't work out however, with his game time being limited due to Rovers having a settled side that was doing well. He returned to Liverpool at the beginning of January, rediscovering his form in the Under 23's.

Clarkson has been loaned to Aberdeen for 2022/23 and made an immediate impact in his debut on 6 August against St Mirren, scoring a stunning goal from thirty yards. He scored four more league goals and was named the club's young player of the season, helping them to a third placed finish. With just one year left on his Liverpool contract speculation linked Clarkson with a move to a number of Championship clubs but he chose instead to join Aberdeen on a longer term basis, signing a four year deal. He featured prominently in 2023/24, scoring four goals in 49 appearances as both a defensive and attacking midfielder. Clarkson continued his development at Aberdeen during the 2024/25 campaign, firmly establishing himself as one of the side’s most reliable midfielders. Wearing the number ten shirt, he became a regular fixture in Barry Robson’s line-up, starting over twenty league games and coming off the bench in a dozen more. His influence was most clearly seen in the Scottish Premiership, where he contributed four goals and an assist, showing an increasing confidence to push forward from midfield and take responsibility in key moments.