In 2014 Bácskay competed at the Gym Festival Trnava where she help Hungary finish fourth in the Youth Team final and individually she finished eighth in the all-around. In 2016 Bácskay competed at the Olympic Hopes Cup where she finished nineteenth in the all-around and eighth in the balance beam finals. Later that year she competed at the Eva Kanyo Cup where she finished fourth among the Espoirs in the all-around and won silver on the vault, gold on uneven bars, and bronze on balance beam and floor exercise. She also competed at the Győr Trophy where she placed seventh in the all-around and second on floor exercise.
2017
In April 2017 Bácskay competed at the Elek Matolay Memorial where she placed first in the all-around, on balance beam, and on floor exercise and she placed second on vault. In May she represented Hungary at International GymSport where she placed fourth in the all-around but won gold on vault. In June she competed at Gym Festival Trnava where she placed sixth in the all-around, second on vault, and third on floor exercise. Later in the summer Bácskay competed at the Slovakian Junior Friendly where Hungary won the gold in the team event. In July she competed at the 2017 European Youth Olympic Festival where Hungary placed fourth in the team final and individually Bácskay placed eighth on vault and fourth on floor. In November she competed at the Olympic Hopes Cup where she placed eighteenth. She ended the year competing at the Hungarian Master Championships where she placed eleventh in the all-around, second on vault, and sixth on floor.
2018
Bácskay started the 2018 season competing at Gymnasiade where she finished seventh in the all-around, second on vault behind Anastasia Bachynska of the Ukraine, and seventh on uneven bars and balance beam. In August Bácskay represented Hungary at the 2018 European Championships. She qualified to the vault final and finished in sixth place. In October Bácskay represented Hungary at the 2018 Youth Olympic Games. She qualified to the all-around and vault finals. In the all-around final she placed 14th. In the vault final Bácskay won the silver medal behind Giorgia Villa of Italy and ahead of Emma Spence of Canada.
Senior
2019
Bácskay turned senior in 2019. She competed at the Hungarian National League Championships in March where she placed first on vault, fourth on uneven bars and floor exercise, and ninth on balance beam. Bácskay made her international debut at the Doha World Cup where she finished 20th in vault qualification, 12th in balance beam qualification, and 23rd in floor exercise qualification. In April Bácskay competed at the 2019 European Championships. During qualifications she placed tenth on vault, becoming the second reserve for the final. She finished thirty-first on balance beam. In September Bácskay competed at the Szombathely World Cup where she qualified to the vault final. During event finals she placed seventh on vault.
