But they squandered the chance to enhance their title prospects with a sloppy display that saw Chris Wood put Burnley ahead after half-time. There was controversy about Wood's goal as it came from a corner that Tottenham felt shouldn't have been given.
But Pochettino's men had a break of their own when Kane, back after missing seven games with an ankle injury, capitalised on a throw-in that was taken in the wrong place. At that stage, Tottenham looked poised to push on for a fifth successive league victory.
However, they paid the price for sloppy defending as Barnes struck with seven minutes left to extend Burnley's unbeaten league run to eight matches. It was a bitter blow for third placed Tottenham, who will be eight points off top spot if second placed Liverpool, currently level on points with City, win at Manchester United on Sunday.
Pochettino was fuming at full-time, confronting referee Mike Dean and his assistants on the pitch in a lengthy rant that showed his frustration at both Burnley's first goal and Tottenham's missed opportunity.