Litex Lovech Despodov was born in
Kresna. He joined
Litex Lovech's Academy at the age of 12 and progressed through the club's youth system. He played an important role in Litex's 2012
BFU U-16 Cup win, scoring two goals in the final against
Cherno More on 26 April. Despodov made his first team debut under
Hristo Stoichkov's management, at 15 years 183 days, in a 5–0
league win over
Kaliakra Kavarna on 12 May 2012, coming on as a substitute for
Svetoslav Todorov. He scored his first goal on 10 August 2013, netting the fifth in a 5–1 away league win over
Pirin Gotse Delchev. On 22 November 2014, Despodov scored his first goal of the 2014–15 season against
Cherno More in a 3–0 away win. On 27 November, he signed his first professional contract with Litex shortly after turning 18. Two days later he scored his second goal of the season in a 2–2 away draw against
Levski Sofia in the 66th minute to clinch a point for his team. Despite the not so good start of the season 2015–16 for Despodov, as he played only in 4 matches from 19 rounds, he became a hero in the quarterfinals of the Bulgarian Cup. In the match against
Levski Sofia, after coming on as a substitute in the 76th minute when the match was 0–0 and Litex was down to 10 people, they eventually ended up winning the game after extra time 3–0, with Despodov scoring 2 of the goals.
CSKA Sofia He joined
CSKA Sofia in the summer of 2016. On 29 April 2017, Despodov scored his first goal in
The Eternal Derby, netting the second in CSKA Sofia's 3–0 win over
Levski Sofia in a
First Professional Football League match. Remaining with the club until January 2019 and becoming popular with the fans of the "redmen", Despodov later began receiving a frosty reception from many of them as a result of becoming part of the ranks of league rivals
Ludogorets Razgrad in 2020.
Cagliari On 30 January 2019, Despodov completed a transfer to
Serie A side
Cagliari, signing a contract until 2023 with an extension option for one more season. CSKA Sofia did not reveal the exact transfer fee, but it was reported that it broke the record for an outgoing
First League transfer of a local Bulgarian player, which until then was
Hristo Stoichkov's
$4.5 million move to
Barcelona in the summer of 1990. He made his debut for the club in a 3–0 away loss to
Milan on 10 February, coming on for
Artur Ioniță in the 75th minute.
Sturm Graz (loan) On 2 September 2019, Despodov was loaned to
Sturm Graz for the 2019–20 season.
Ludogorets Razgrad On 5 October 2020, Cagliari announced that Despodov would be loaned to
Ludogorets Razgrad with an option to buy. The deal would see him remain at the club for the duration of the
2020–21 campaign. On 19 June 2021, Ludogorets announced that he had signed permanently with the club. On 12 March 2023, Despodov scored his first hat-trick for Ludogorets in a 3–2 home win over
Cherno More in a league match.
PAOK On 4 September 2023,
Super League Greece club
PAOK announced the signing of Despodov on a four-year contract, for a reported fee of €3 million. On 28 January 2024, he scored a brace in a home match against
Panathinaikos, securing a 2–1 win. ==International career==