[5] Taken to the ice rink by his father, Kovtun began skating at age four in Yekaterinburg and was coached mainly by Maria Voitsekhovskaia in his early years.

He debuted on the ISU Junior Grand Prix (JGP) circuit in the 2011–12 season, winning gold at his first event in Romania and then silver in Estonia.

[9] Kovtun qualified for the JGP Final in Sochi, Russia, where he won gold by eleven points over silver medalist Joshua Farris.

[13] In his European debut, Kovtun placed seventh in the short program, fourth in the free skate, and fifth overall with a total score of 226.57 points.

On Thursday 13 February 2014, Plushenko decided to withdraw from the men’s singles competition following his pre-short program warm up during which he sustained a back injury.

[14][17] He placed first in both segments to win the gold medal in China, ahead of the Olympic champion, Yuzuru Hanyu, and Richard Dornbush.

At the 2015 European Championships, he ranked fourth in the short program and second in the free skate, ending in second place overall, behind Javier Fernández.

Due to the November 2015 Paris attacks, the free skate was cancelled and the SP standings were deemed the final results.

In late January, Kovtun was awarded the bronze medal at the 2016 European Championships in Bratislava, having placed second in the short and sixth in the free.

[21] After taking the bronze medal at the 2016 CS Finlandia Trophy, Kovtun finished seventh at both of his Grand Prix assignments, the 2016 Skate America and 2016 Cup of China.

[22] He won the silver medal at the 2017 European Championships, earning new personal bests in the short program, free skate and overall.

Kovtun withdrew from the 2017 Finlandia Trophy and his first Grand Prix assignment, 2017 Skate Canada, due to a back injury.

[23] Kovtun competed in the short program at 2017 Skate America, placing twelfth, and withdrew as a result of continued back and knee problems.

Commenting afterward on his disappointing result, Kovtun said it was a "novel experience" and that he "was physically unable to train, simply because I could not bend down to tie my laces and then straighten up again.

[25] Coached by Elena Buianova and Alexander Uspenski at CSKA Moscow,[25] Kovtun began his season by winning the gold medal at the 2018 CS Tallinn Trophy.