Mozgov began skating in 1999. His parents introduced him to skating to improve his health. Early in his career, he competed with Sabina Adigamova.
Mozgov and Evgenia Kosigina made their international debut in the 2008–09 season. The following season, they received a Junior Grand Prix (JGP) assignment in Hungary, where they placed fifth. He ended their partnership after they finished 12th at the 2010 Russian Junior Championships.
In the 2010–11 season, Mozgov competed with Tatiana Baturintseva. They received no JGP assignments and parted ways after placing 9th at the 2011 Russian Junior Championships.
Mozgov teamed up with Anna Yanovskaya in 2011. They were coached mainly by Svetlana Alexeeva at the Medvedkovo rink in Moscow.
Yanovskaya/Mozgov won the bronze medal at their first Junior Grand Prix event, in Gdańsk, Poland, and then gold in Tallinn, Estonia. Their placements qualified them for the Junior Grand Prix Final, where they placed second in the short dance, third in the free, and took the silver medal ahead of Alexandra Stepanova / Ivan Bukin. After winning the gold medal at the 2012 Winter Youth Olympics, they placed fourth at the 2012 Russian Junior Championships. At the 2012 World Junior Championships, they were third in the short dance. During the free dance the referee stopped their music because Mozgov's left bootstrap had come loose. Yanovskaya/Mozgov finished fourth overall behind American ice dancers Alexandra Aldridge / Daniel Eaton who moved up the rankings and took the bronze medal.
In 2012–13, Yanovskaya/Mozgov won a pair of silver medals at JGP events in Austria and Slovenia and qualified for the JGP Final in Sochi, Russia, where they finished fourth. They won the bronze medal at the 2013 Russian Junior Championships.
In 2013–14, Yanovskaya/Mozgov began their season by taking gold at the 2013 JGP Slovakia in Košice. They won another gold at the 2013 JGP Estonia, qualifying them for their third JGP Final in Fukuoka, Japan. Setting personal bests, Yanovskaya/Mozgov placed first in both segments at the final and won the gold medal ahead of Kaitlin Hawayek / Jean-Luc Baker. After placing second to Stepanova/Bukin at the 2014 Russian Junior Championships, they took the silver medal at the 2014 World Junior Championships in Sofia, finishing second to Hawayek/Baker.
Yanovskaya/Mozgov decided to remain in the junior ranks in the 2014–15 season. In addition to Moscow, they also trained in Liepāja in the summer. Mozgov recovered from a knee injury early in the season. During in the 2014–15 JGP series, the duo won gold medals in Estonia and Croatia, earning qualification to their fourth JGP Final. At the event, held in December in Barcelona, they outscored Alla Loboda / Pavel Drozd for the gold and stood atop the podium for the second consecutive year. In March, they competed at the 2016 World Junior Championships in Tallinn, Estonia. Ranked first in both segments, they were awarded the gold medal ahead of Lorraine McNamara / Quinn Carpenter.
Although still age-eligible for junior events in the 2015–16 season, Yanovskaya/Mozgov decided to move up to the senior ranks. Debuting on the Grand Prix, they placed sixth at the 2015 Skate America and 2015 Trophée Éric Bompard. In December, they finished sixth at the 2016 Russian Championships in Yekaterinburg. On 2 May 2016, the Russian media reported that their partnership had ended.
According to the Russian media, Mozgov has teamed up with Betina Popova.
GP: Grand Prix; CS: Challenger Series; JGP: Junior Grand Prix
Small medals for short and free programs awarded only at ISU Championships. At team events, medals awarded for team results only.