Allardyce became England manager in July 2016 but was sacked by the FA after 67 days for behaving "inappropriately" when caught up in a newspaper sting. "We want to get off to winning ways", he explained.
Kane further revealed that England, who are famous for their bad record in shootouts, would look to play the attacking game and focus more on scoring.
Southgate was speaking as he waved off Martin Atkinson from St George's Park ahead of the Premier League referee's charity bike ride to Russian Federation.
Kane, who jetted off to Russian Federation with his team-mates on Tuesday, was still at school when Pochettino fouled Michael Owen in the 2002 World Cup, allowing David Beckham to score from the penalty spot and earn England a 1-0 group-stage win over Argentina.
"I said to all of the team before the game: 'I expect to see mistakes tonight and if I don't there'd probably be a problem because we won't be trying to be as good as we might'. When I decide, 'Okay, we are not going with that one but we will have a think about this one, ' they really respect that". "But don't worry, we are going to get there'".
England wrapped up their St George's Park camp preparations by staging an internal match on the Sir Bobby Charlton Pitch.
The player will conduct separate preparatory work at the team hotel.
We have the talent in this squad to win football matches, it's all a case of whether Harry Kane and co can do it on the biggest stage in the game. You've got your minutes.
For those of you who are of a cynical nature, you may well point out that a different England player gives a similar sounding interview before every major tournament, and in all honesty we won't argue as you'd be pretty much spot on.
"But a couple of them went flying into tackles and you're thinking "bloody hell", and you're just jumping out the way of things".
"One night I was playing golf - next day they said you can't carry on". Actually at that point I'm not convinced they do.