In July 2014, he was signed by West Ham, who have loaned him to Huddersfield Town, Milton Keynes Dons and Charlton.

Having started his career at Charlton Athletic since joining the club's U13 side, Poyet was offered a scholarships by The Addicks in March 2011 and signed his first professional deal five months later.

[6][7] Poyet made his professional debut on 21 January 2014 in a 3–0 FA Cup third-round win over Oxford United, entering the match as an 81st-minute substitute for Dale Stephens.

[15][16] On 26 June 2014, Charlton announced that Poyet had indicated that he would not be signing an extension to his contract ending on 1 July 2014, and would be exploring his options elsewhere.

[19][20] His father, Gus, manager of Sunderland, was also keen to sign his son, but backed away, citing fear of nepotism which could have caused conflict in the dressing room.

[23] On 23 August 2014, Poyet made his debut for West Ham United, against Crystal Palace, coming on in the 83rd minute for Mauro Zárate in a 3–1 away win.

[24] Four days later, he started for the first time, playing the full 120 minutes and scoring in the penalty shoot-out as the club lost at home to Sheffield United in the second round of the League Cup.

[26] Following his loan spell at Huddersfield Town came to an end, he made his first appearance for West Ham United in the FA Cup third round match against Everton, coming on as a 68th-minute substitute, in a 1–1 draw on 6 January 2015.

On 23 July 2015, as a substitute for captain Kevin Nolan, Poyet scored the winner in a penalty shootout at the Ta' Qali National Stadium in Malta, at the end of a 1–1 aggregate draw with Birkirkara in the second qualifying round of that season's UEFA Europa League.

[35] He made his début on 22 November at the John Smith's Stadium, playing the full 90 minutes in a goalless draw against Sheffield Wednesday.

[36] He played the next game against Bolton Wanderers, but was not included in their 2–1 win over Brentford, and returned to West Ham at the end of his loan.

[42][43][44] His loan spell with Milton Keynes Dons was cut short in January 2016 by manager Karl Robinson.

[46] Five days later on 9 January 2016, he played the first game of that loan, featuring for the entirety of a 2–1 loss at Colchester United in the FA Cup.

[57] Poyet made his debut for the club, starting the whole game, in a 3–1 win against Sport Boys Warnes on 28 April 2017 in the Copa Libertadores.

[89] In February 2015, the Uruguayan Football Association announced their intention to call up Poyet for two games in March 2015 against Portugal and Uzbekistan ahead of the 2015 FIFA U-20 World Cup.

[90] In March 2015, he made his debut for Uruguay U20, coming on as a substitute for Mauro Arambarri in a 1–1 draw against France U20 at Clairefontaine.

[93][94] Poyet made his first start of the tournament against Mali U20 on 6 June 2015, as Uruguay U20 drew 1–1 and progressed to the knockout stage.

[95] He, once again, started in the tournament against Brazil U20 in the Round of 16, playing 120 minutes and was one of the four players to successfully convert the spot kick in the penalty shootout, as the U20 side lost 5–4.