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 made his debut for Liverpool in the 5-0 defeat in the quarter-finals of the Carabao Cup against Aston Villa on 17 December 2019.
On 9th 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 16th August 2021 a season long loan with his boyhood club 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 6th August against St Mirren, scoring a stunning goal from thirty yards.