[5][6] Addo-Carr was born in Blacktown, New South Wales, Australia, and is of Aboriginal descent from Gunggandji, Birrbay and Wiradjuri people.

[8] Addo-Carr attended Matraville Sports High School and played his junior rugby league for the La Perouse Panthers and Moore Park.

In 2012, Addo-Carr played with the South Sydney Rabbitohs SG Ball team before being released due to an off-field incident.

[17] Addo-Carr made his Melbourne Storm debut in round 1 of the season against the Canterbury-Bankstown Bulldogs at Belmore Sports Ground.

[18] In Round 3 against the Brisbane Broncos, Addo-Carr scored his first club try for Melbourne in the final minutes to win the match 14–12 at AAMI Park.

[25] In the 2018 NRL Grand Final, Carr scored an 85-metre intercept try in the 21–6 loss to the Sydney Roosters, bringing his year tally to 18 tries from 25 games.

Before the start of the 2019 State of Origin series, Addo-Carr was one of the New South Wales players who declared they would not sing the Australian National Anthem.

On 27 April, Addo-Carr was fined $1000 by the NRL and placed under investigation after he broke strict lockdown protocols with fellow player Latrell Mitchell by going on a weekend camping trip during the Coronavirus pandemic.

In scoring six tries in a single game, Addo-Carr set a new Melbourne Storm club record and became the first player in Australia in 71 years to do so; the last was Jack Troy in 1950 while playing for Newtown.

[35] Addo-Carr played a total of 22 games for Melbourne in the 2021 NRL season and scored 23 tries as the club won 19 matches in a row and claimed the Minor Premiership.

[43] Addo-Carr was selected in the 24-man squad for the Australian Kangaroos by Mal Meninga to travel to England to compete in the 2021 Rugby League World Cup.

[44] He proved the critics wrong, as journalist Paul Crawley said in response to Brad Fittler's controversial decision to not select him for the 2022 State of Origin series: "so from a Canterbury recruitment perspective and Gus is talking about having all this money that is available in 2024, what rep player or future rep player with half a clue would want to go to the Bulldogs knowing it could be the end of your representative career if you go there?

Addo-Carr played on the wing against Fiji Bati at Emerald Headingley Stadium in Leeds scoring a double as the Kangaroos won 42-8.

After the game, the media posed serious questions to Fittler for making the call not to pick Addo-Carr for the Blues in 2022, in a series they would ultimately lose.

[47] After scoring four tries against Scotland including a "world class" try in the final minute, the "Foxx" was labelled a star, unstoppable and a "freaky man".

The following day it was confirmed that Addo-Carr would require surgery and was ruled out for eight weeks, adding to an already extensive injury list for Canterbury.

Addo-Carr did not contest a $682 fine and a three-month suspension from driving (without having a conviction recorded) after testing positive to cocaine, but claimed that he did not knowingly take any drugs.

[72] On 31 October, Addo-Carr was sacked by the Bulldogs due to breaches of his playing contract and lack of transparency when he faced the board.