While Cenk Tosun offered flashes of promise on his Everton debut, in the end he was powerless to stop himself from becoming the latest striker to be left in the shade by Harry Kane. Tosun’s muscular cameo was an isolated bright spot for Sam Allardyce, who was shocked by his team’s miserable capitulation in the second half, but a resigned chuckle was the best Everton’s manager could muster after being asked about Kane’s poaching masterclass. “He’s very good,” Allardyce said.

That was an understatement. Kane ended 2017 with the numbers to show he is the most potent attacker in Europe and he shattered another statistic in this thumping victory for Mauricio Pochettino’s side, scoring twice to break Teddy Sheringham’s record of 97 Premier League goals for Tottenham.

Kane has a long way to go to surpass Jimmy Greaves as the club’s all-time leading scorer with 266 goals but worryingly for opposition defences, he has no intention of slowing down after becoming the first player in the top division to reach 20 goals this season. “It’s something I’m very proud of but it’s on to the next one,” he said. “We’ve got to keep going.”

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That killer mentality explains why the 24-year-old has succeeded in transforming himself into one of the world’s elite players and Pochettino was effusive in his praise when he was asked if Kane can break Andy Cole and Alan Shearer’s joint record of 34 goals in a single Premier League season. “It is a lot of numbers,” Tottenham’s manager said. “He can do everything. He has the potential to achieve what he wants. He is always thinking and trying to improve.”

Kane was not without assistance in a romp that moved Tottenham closer to Chelsea, Liverpool and Manchester United in the fight to qualify for the Champions League. Son Heung-min also sparkled, creating Kane’s first with an electric burst two minutes after half-time, and the South Korean’s opener in the first half was also a significant moment, allowing him to emulate Jermain Defoe by becoming the second Tottenham player to score in five consecutive home games in the Premier League.

Facebook Twitter Pinterest Christian Eriksen finishes a dazzling team move to put Tottenham 4-0 ahead. Photograph: Alex James/JMP/Rex/Shutterstock

The indignities for Everton piled high. Allardyce accepted some culpability for his part in this chastening defeat, admitting he picked too many offensive players, but he was more troubled by his team’s collapse after a sprightly offering in the first half.

“The gulf was massive,” Allardyce said. “It was a shock. I saw some of the good sides last week against Liverpool and I saw some of the worst sides today. It’s worrying how bad our performances have become in such a short space of time. I didn’t expect to see it and I have to address it as quickly as possible.”

Everton could not be accused of a lack of ambition during the early stages. Failing to react after selling Romelu Lukaku last summer has been a major factor in their muddled campaign but the arrival of Tosun from Besiktas has infused them with fresh hope. The £25m forward was prolific for his former club and Everton could take encouragement from his speed and movement, which troubled Davinson Sánchez at times. “With better service hopefully he’ll do what he did for Besiktas,” Allardyce said.

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Tosun created an early chance for Wayne Rooney after holding the ball up well and his near-post flick from Gylfi Sigurdsson’s corner almost led to Everton taking a shock lead. However, Rooney was just offside as he headed past Hugo Lloris from close range and that scare stung Tottenham into action.

Everton were too open down their left side, with Sigurdsson struggling to track Serge Aurier’s raids, and Christian Eriksen took note of that weakness, finding the buccaneering right-back with a superb, raking pass. Aurier’s delivery can be patchy but here he drove a low centre into the middle, enabling Son to take advantage of poor marking by tapping past Jordan Pickford.

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“We have talked about Son’s performances for four months,” Pochettino said. “He is more mature than last season.” In the 47th minute Son demonstrated his dribbling ability by twirling away from Jonjoe Kenny, rendering the inexperienced right-back an irrelevance as the move unfolded. Mason Holgate was unable to stop him from fizzing the ball into the middle, where Kane was waiting.

Allardyce felt that Kane was offside but Everton were brutally exposed. Out of the FA Cup and winless in the league since 18 December, their season is drifting aimlessly towards an unsatisfactory conclusion, and Kane had already scored his second – guiding home Eric Dier’s low centre – by the time Tosun made way for Dominic Calvert-Lewin in the 62nd minute.

Rooney’s late booking for an ugly chop on Jan Vertonghen encapsulated Everton’s frustration and Tottenham finished with a stunning fourth goal, Dele Alli’s lovely backheel teeing up Eriksen for an emphatic shot past Pickford. “Maybe I’ll go back to being a bit more boring,” Allardyce said. Tottenham have no such worries.