As a member of the Slovak national ice hockey team, she represented Slovakia at the 2010 Winter Olympics and at six IIHF Women's World Championship tournaments.

Lucák Čupková made her senior club debut with ŽHK Šarišanka Prešov in the 2004–05 season of the Slovak Women's Extraliga.

She played with ŽHK Šarišanka Prešov through the 2007–08 season, with the exception of fourteen games spent with the Popradské Líšky (lit.

In two games with ŽHK Michalovce during the 2022–23 Slovak Women's Extraliga season, Lucák Čupková tallied two goals and six points.

As a junior player with the Slovak national under-18 team, Lucák Čupková competed in two IIHF World Women's U18 Championships, winning Division I bronze medals in the 2009 and 2010.