Abu Kamara joined Norwich City's Under-18s scholars for the 2019/20 season. He had joined Norwich at Under-11 level and .played for Norwich's Under-16s in their 1-0 win over Ipswich in the Hospital Cup (8th May 2018)

He hit the headlines when he scored a hat-trick for Norwich's Under-18s as they beat Wolves 4-2 in the Midlands on 18th December 2020 in the third round of the FA Youth Cup.

On 13th January 2021 he signed his first professional contract with Norwich, keeping him at City until summer 2023. He said “It’s a proud moment for me and my family, it’s every kid’s dream to become a professional footballer and especially at a club like this, it’s a massive achievement for me and my family, so I’m over the moon. The club have shown a lot of faith in me and that gives me more confidence to do well, I really appreciate it from them. I try to get as many goals as I can to repay their faith in me. I’ve been here roughly six or seven years, I joined here at the end of the under-11s, and I’ve enjoyed every minute of it. From the school programme to where I am now, it’s been amazing really.”

Kamara was on the bench for Norwich's 0-0 draw with Brighton on 2nd April 2022.

He finished the 2021/22 season as the leading scorer in the Premier League 2 with 15 goals. He is described as bringing "composure, intelligence and quality to a team."

After being on the bench for a couple of Championship matches in 2022/23, Kamara made his first team debut on 1st April 2023 coming on as substitute for Teemu Pukki with 20 minutes remaining in the 1-0 home defeat to Sheffield United. He finished the season with three appearances. He also scored 13 goals in 15 Premier League 2 appearances for the Norwich Under-21s

On 21st June 2023 it was reported by Football Insider that Brentford were interesting in signing Kamara who is said to be firmly on their radar.

On 19th July 2023 he signed for Portsmouth on a season-long loan. "Abu is an exciting young prospect who can play across the front line," said Pompey boss John Mousinho. "He's got a really good goalscoring record for Norwich's development side. Now he comes here and will be competing for a place at League One level."

In January 2024, there was speculation that Championship sides Leicester and Stoke were both keen on signing Kamara who had scored seven goals in 34 appearances for Portsmouth as they pushed for promotion. With Kamara having entered the final 18 months of his Norwich deal, although there is a one-year option in the Canaries' favour, clubs are monitoring his situation in the final days of the January transfer window, TeamTalk have claimed. 

On 16th April 2024 Kamara was in the Portsmouth team that beat Barnsley 3-2 to secure the League One title. He played 52 games scoring 10 goals.

On 24th May 2024, Kamara was called up to the England Men's Elite Squad (formerly known as the Under-20s) for friendlies with Sweden and the Republic of Ireland in June 2024. Teammates Kellen Fisher and Brad Hills were also called up as was former Canary Alex Matos. He started for England Under-20s as they beat Sweden 2-1 on 7th June. Kamara played 66 minutes before being replaced by Millwall winger Romain Esse.

On 17th July 2024 it was reported that Portsmouth were readying an official bid for Kamara. The south-coast side have been open about their desire to reunite with Kamara after an impressive season on loan at Fratton Park. It's understood that there's been no contact between the clubs at this stage. Football Insider's Pete O'Rourke says Pompey will test the Canaries' resolve, however, and try to tempt them into selling with an official offer. During the summer of 2024, Abu was also linked with moves to Southamptom and Hull City. In early August, Celtic were also reported to be watching

Kamara scored his first goal for Norwich's first team as he opened the scoring in their 4-3 win at home against Stevenage in the League Cup on 13th August 2024. After playing in the first two league games of the 2024/25 season as substitute, he submitted a transfer request. Hull, Luton and Oxford all showed an interest. On 30th August 2024 he completed a move to Hull City for an undisclosed fee, believed to be a package in excess of £5m with add ons, plus a sell on fee.