Complex sentence: Liverpool finished no higher than sixth in the Premier League during the next two seasons and thus failed to qualify for European competition until the 1995--96 season when they entered the UEFA Cup, but again progressed no further than the second round.
Split into simpler sentences: Liverpool finished no higher than sixth in the Premier League during the next two seasons, and thus failed to qualify for European competition. In the 1995--96 season they entered the UEFA Cup, but again progressed no further than the second round.