In March 2012, it was announced that Carlton Morris, a centre-forward from Cambridge, would be joining the Canaries as an Academy Scholar in the summer of 2012. He had first joined Norwich at the age of 11.

He played a key role in both legs of the FA Youth Cup final which City won 1-0 and 3-1 against Chelsea in April/May 2013.He made 25 appearances for the Under-18s in 2012/13 scoring seven goals.

He was called up to the England Under-19 squad in August 2013. He made his England Under-19 debut against Estonia on 5th September 2013, coming on as a substitute at half-time and scoring 20 minutes later in a 6-1 win in Tallinin. Norwich teammates Harry Toffolo and Josh Murphy also played.

He signed a professional contract with the Canaries on 16th December 2013.

On 5th August 2014 Morris was loaned to Oxford United on a six month loan and made his debut on 8th August 2014 in the 1-0 home defeat to Burton Albion. He made seven starts for Oxford, scoring once but found opportunities limited and returned to Carrow Road on 23rd October 2014.

On 27th November 2014, Morris was loaned to York City until 4th January 2015. He made his York debut in the 1-1 draw at Plymouth on 29th November, coming on as a 77th minute substitute. He made eight appearances for York.

He made his Norwich debut coming on as substitute in the 1-0 home defeat to Wigan on 4th March 2015.

On 13th July 2015, he was loaned to Scottish side Hamilton where he scored eight goals in 33 appearances.

On 31st January 2017, he was loaned to Championship rivals Rotherham until the end of the 2016/17 season. A hamstring injury delayed his debut for two months. He played on 1st April 2017 when a 1-0 home defeat to Fulham saw the Millers relegated to League One.

Morris signed for Shrewsbury on loan on 11th July 2017 until the end of the 2017/18 season. He made 54 appearances, scoring 10 goals, as Shrewsbury reached the League One play-off finals and the Checkatrade Trophy final.

Injuries plagued much of Morris' 2018-19 season but he extended his contract until 2021 on 22nd May 2019 before returining to Rotherham on a season long loan. This was cut short on 8th January 2020 despite him making 28 appearances and scoring four goals for the Millers. He then immediately joined Russell Martin's MK Dons on loan until the end of the 2019/20 season. He scored on his debut on 11th January in a 1-1 draw at Coventry. He made a total of 10 appearances scoring twive before the season was ended due to Coronavirus.

On 24th July 2020 he rejoined the MK Dons on loan for the 2020/21 season. After scoring five goals in 26 appearances across the two loans, he was linked with a move to Championship side Barnsley in the January 2021 transfer window which could allow Norwich to gain some income from the transfer. The loan with MK Dons was ended at the beginning of January 2021. He scored four goals in 24 appearances. He joined Barnsley on 7th January 2021 for a fee reported to be £300,000. He signed a two and a half year deal.

Barnsley were relegated from the Championship on 22nd April 2022. He made 30 appearances scoring nine goals.

Morris moved to Luton on 7th July 2022 for an undisclosed fee, signing a long term deal.

On 27th May 2023 Morris scored Luton's first penalty in the shoot-out to determine the Championship play-offs. Luton won 6-5 on penalties after the game had finished 1-1. Luton were promoted to the Premier League.

Morris and Luton were relegated from the Premier League on 19th May 2024 in a 4-2 home defeat to Fulham in which Morris scored his 11 goal of the season (playing 43 games)