He played little part in the Aston Villa teams which came second in the 1993 Premier League and won the 1994 Football League Cup, but played a more active part in Sunderland's promotion to the Premier League in 1996. Kubicki was controversially dropped from the team by manager Peter Reid for a game against Derby County when selecting Kubicki would have meant him overtaking the club record for consecutive appearances in the post-war period, still held by George Mulhall.
He later began a coaching career and coached Legia Warsaw and Polonia Warsaw.
Kubicki is now a coach of Polish second division team Lechia Gdańsk. In October 2007, Kubicki was arrested by the police due to a bribery scandal involving the sale of one of the Warsaw sport centers. Due to this scandal, Kubicki has been suspended as coach of Lechia Gdańsk.