Mura won his first senior Grand Prix medal, gold, at the 2012 Trophée Éric Bompard.
Mura finished sixth at the Japanese Nationals and was assigned to the 2014 Four Continents Championships where he won the gold.
His next assignment was the 2014 NHK Trophy, where he placed third overall, qualifying for the 2014–15 Grand Prix of Figure Skating Final, in Barcelona, Spain.
At the Grand Prix Final, Mura finished last in the short program and fourth in the free skate, ending fifth overall.
In October 2024 Mura joined forces with Olympic champion Shizuka Arakawa, Olympic bronze medalist Daisuke Takahashi, Kana Muramoto, Kazuki Tomono, Keiji Tanaka, Yuna Aoki, Kosho Oshima, Yuto Kishina and Rena Uezono to launch the members-only official fan community F-Ske on the platform FANICON.