Born in Oldham, Kent joined Liverpool's Academy at the age of just seven in 2003. He progressed through the ranks to become a reserves regular in 2014/15, when he was rewarded with a four and a half year contract soon after his eighteenth birthday. After featuring in Liverpool's pre-season tour of the Far East and Australia, Kent was loaned to Coventry City for the first half of 2015/16. In the winger's final appearance before returning to Liverpool, Coventry beat Crewe 5-0 away from home and Kent was given a standing ovation by his teammates when he returned to the dressing room. 

Kent made his Liverpool debut away to Exeter in the 3rd round of the FA Cup on 8 January 2016 and after being an unused substitute in the second leg did not feature again for the first team that season. Kent spent the whole of 2016/17 with Barnsley in the Championship, being voted their young player of the season as he scored three times in 44 league appearances. Kent signed a new long-term contract and then joined Freiburg on loan in the Bundesliga in the summer transfer window of 2017, making six appearances before returning to Liverpool in January. Kent spent the second half of the season on loan at Bristol City in the Championship, playing ten times in the league.

On 22 July 2018 he was loaned to Steven Gerrard's Glasgow Rangers for the 2018/19 season. He had an impressive season there, creating the only goal at Ibrox as Rangers beat Celtic for the first time in six years and also scoring in a 2-1 defeat in the Old Firm game at Parkhead. He went on to win the Scottish PFA Young Player of the Year Award at the end of the season. Despite this, the quality of the players ahead of him at Anfield meant he remained some way of the Liverpool first team and at the last moment before the Scottish transfer window closed on 2 September he signed for Rangers for an undisclosed fee.

Kent continued to impress in 2019/20 and was linked with a move back to England in September 2020, with Leeds a possible destination. However, he reaffirmed his commitment to Rangers, saying he wanted to remain and further develop under Steven Gerrard to win trophies with the club. He scored ten goals in 37 appearances as Rangers won the title for the first time in ten years and the following season was a key member of the side that exceeded all expectations by reaching the Europa League final, losing on penalties to Eintracht Frankfurt. 2022/23 was more of a mixed season for Kent and on the expiration of his contract he signed for Turkish side Fenerbahce. He scored on his debut in a 5-0 victory over Zimbru Chisinau in a Europa Conference League qualifier. It was the Conference League where he got the most minutes on the pitch in his first season there, appearing ten times compared to just eight in the league. The appointment of Jose Mourinho as manager in the summer of 2024 didn't increase Kent's playing chances and he was forced to train alone. In October 2024 his contract was terminated by mutual consent. In April 2025, Kent agreed to sign for MLS side Seattle Sounders for the duration of the season, with an option to extend for 2026.