The win moved Spurs a point and a place above Liverpool into third with 61 points from 30 games and restored a five-point gap over fifth-placed London rivals Chelsea who they visit on April 1 in the battle for the top four Champions League spots.
The loss of England global Kane, who went off with an ankle injury on the half-hour, took some of the gloss off a hard-won comeback victory after the disappointment of losing to Juventus in the Champions League last 16 in midweek.
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"We hope it's not a massive issue for him and hope he is available for the team as soon as possible".
However, the loss of Kane to an ankle injury presents a major concern to Pochettino who will be anxious to learn the full extent of the forward's problem having twice lost the player to a similar injury last season.
England's opening game is against Tunisia in Volgograd on June 18th.
With 30 minutes gone, Kane got on the end of a lofted Christian Eriksen pass and finished past Asmir Begovic before colliding with the Cherries goalkeeper.
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I've never seen so much hard work and passion from so dedicated a group of people . "I'm a huge fan of them". Other schoolchildren in the crowd were equally thrilled to meet the royal couple and shake their hands.
But two goals from Son Heung-min in between strikes from Dele Alli and Serge Aurier ensured Spurs would run out comfortable winners.
He volleyed in Alli's cross at the back post in the 62nd, before displaying his composure by rounding Begovic to seal three points in the 87th.
But the Bournemouth "keeper was at fault as he parried Kieran Trippier's cross into Aurier's path, and the Ivorian headed home Spurs" fourth in stoppage time.
"It is an unbelievable personal milestone", said Cech who had waited since December for a shutout. His cross for Mustafi's opener made him the fastest man to 50 Premier League assists, taking 141 games.
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The Rockets were 1 for 12 from 3-point range before Harden drained back-to-back shots with 2:48 left in the third. Chris Paul finished with 14 points for Houston but missed a go-ahead three-pointer with 14.5 seconds to play.
Tottenham tried to find their way back into the match and thought they had did just that on the half hour mark when Harry Kane found the back of the net, but his goal was not given because he was adjudged to have been in an off-side position.