Goals from Erling Haaland's first-half strike secured the victory for Pep Guardiola's side, leaving them top of the Premier League table ahead of Arsenal.
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But the result means Scott Parker's Burnley side have just 20 points with four games remaining, an insurmountable deficit to safety.
The Clarets are 13 points from safety as Tottenham Hotspur currently occupy 18th place with 31 points, with West Ham narrowly out of the relegation zone on 33 points.
More follows.