Tottenham are not making things easy in their quest for a Premier League top-four finish. On a madcap afternoon at Bournemouth, they played for 42 minutes with nine men, after having Son Heung-min and the substitute Juan Foyth sent off, while they were fortunate not to lose Eric Dier to another red card. Their rap sheet also featured five yellow cards.

Mauricio Pochettino’s depleted team held out until stoppage-time and it had to be said that Bournemouth were strangely toothless opposition, not entirely comfortable when the onus was on them to force the issue. But they found the killer blow when Nathan Aké showed more desire than anybody around him to head home from Ryan Fraser’s corner.

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The upshot for Spurs was that unless results involving the teams immediately below them go in their favour on Sunday, they will need something – possibly a victory – at home to Everton on the final day to ensure a return to the Champions League via league position.

Spurs’ poor form goes on. They have now lost seven of their past 11 league fixtures and this was not the preparation they had in mind for the second leg of their Champions League semi-final at Ajax on Wednesday, in which they trail 1-0.

Pochettino’s players were the architects of their own downfall, with Son getting himself into trouble when he lost his cool on 43 minutes to push Jefferson Lerma, and Foyth leaping into a poor challenge on Jack Simpson almost immediately after his introduction as a half-time substitute.

Bournemouth played like a mid-table team at the end of their season, although Eddie Howe could relish a nerveless professional debut from the 19-year-old goalkeeper Mark Travers, but at least they could enjoy the finale. The irony that Aké’s goal came from a set piece, when Spurs had the personnel to defend properly, was not lost on Pochettino. Bournemouth are now one point shy of their best Premier League points tally.

Facebook Twitter Pinterest Son Heung-min shoves Jefferson Lerma in the first half. The South Korea forward was shown a straight red card. Photograph: Mark Leech/Offside/Getty Images

Howe felt a little cheeky, to borrow the word that he used, in pointing out Dier should have been the first man sent off but everybody inside the stadium could see it – apart from the referee, Craig Pawson. Already booked for taking out Fraser on the break, he slid back to trip Joshua King on another Bournemouth counter in the 34th minute.

The entire Bournemouth team moved towards Pawson to demand Dier’s dismissal and, when he allowed play to continue, they were almost caught cold when Spurs broke through Dele Alli. Pawson had the whistle to his lips before he appeared to think better.

Dier would get away with another rash challenge on 42 minutes when he went into Callum Wilson inside the area. Dier got none of the ball and all of the man, and it was a mystery as to how Bournemouth did not get a penalty. Pochettino withdrew Dier at the interval because he had become a danger to his team and he did likewise with Toby Alderweireld because he was also on a yellow card.

The Tottenham manager said he did not want any more dismissals but Foyth would make a mockery of that hope after just three minutes. The young defender over-ran the ball and, stretching to rectify the situation, his momentum took him into Simpson. The Argentinian’s boot was extremely high.

Son had felt the red mist descend after conceding a free-kick to Fraser. He picked himself up, felt Lerma aim a dig in his back and raised his hands in an aggressive manner. Lerma went down dramatically but Son ought to know better. It was possible to wonder whether Pawson’s lack of action over Dier had preyed on the referee’s mind.

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Prior to the dismissals, the central figure had been Travers. He was almost lobbed from the halfway line by Lucas Moura in the early running but subsequently made a string of saves in the opening 33 minutes, when Spurs were in the ascendancy. The best was a flying tip over the bar to keep out an Alli header. Son also blew a good chance on 31 minutes when he fired high after running around Steve Cook.

King was denied by Hugo Lloris in the 38th minute – the goalkeeper reached the ball before taking out the man in the area – while Simpson went close in first-half stoppage-time. But with the two-man superiority, Bournemouth hardly laid siege to the Spurs goal. Jordon Ibe drew a routine save from Lloris with a shot from distance and the substitute Lys Mousset blew a free header.

“Of course we’re not happy,” Pochettino said. “We dominated the first half and we played for so long with nine men and their goalkeeper is the man of the match. Crazy. We’ve now got two finals against Ajax and Everton.”