[3] Despodov made his first team debut under Hristo Stoichkov's management,[4] 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.

[10] 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.

[11] 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.

[18] On 4 September 2023, Super League Greece club PAOK announced the signing of Despodov on a four-year contract,[19] for a reported fee of €3 million.

On 7 February 2015, Despodov made his first appearance for Bulgaria, in a goalless draw with Romania in a non-official friendly match, coming on as a late substitute for Stanislav Manolev.

[22] Despodov finally made his complete debut for Bulgaria on 23 March 2018 in a friendly match against Bosnia and Herzegovina, replacing Spas Delev shortly after the one hour mark.

On 29 March 2022, Despodov opened the scoring in the friendly match against Croatia, but was dismissed from the field almost immediately after that due to receiving a second yellow card for taking off his shirt as part of his goal celebration.