Mauricio Pochettino's men have finished third and second in the previous two Premier League campaigns, but have failed to make an impression in European and domestic cup competitions.
Spurs crashed out of the Champions League and Europa League last year at Wembley, their temporary home for the 2017/18 season.
Tottenham's Wembley woe continued on Sunday after a Marcos Alonso double sealed the first three points at the national stadium for Chelsea.
Speaking after the game, Kane highlighted the importance of winning a trophy this term. The striker said, "Yes, that is what we have to do.
"Third, second. The only way we can improve is to win something or win the league. That is our aim. That is always our aim. We have to keep doing what we are doing and hopefully it will happen.
"It is another year that we have been playing with each other. Another year of getting used to the way that we play with the manager."