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Burnley's revival under caretaker boss Mike Jackson continued as a thrilling 2-1 win at Watford pulled them five points clear of the relegation zone.
Manchester United's troubles continued as a 1-1 draw with Southampton further damaged their top-four hopes.
Spurs beat Norwich City 3-0 at home, while Ralf Rangnick made a winning start as Manchester United's interim manager with a 1-0 victory over Crystal Palace.
Norwich sacked Farke this month when Smith was also fired by Villa following a run of five straight league defeats.
The 32-year old from Harlow, England, has played for Scottish giants Celtic and Norwich City in the Premier League.
The result lifted Watford two places to 18th on eight points from 12 games while promoted Norwich, who last won in September when they upset champions Manchester City 3-2 at Carrow Road, dropped to the foot of the table on seven points.
Juergen Klopp’s side, runners-up to Manchester City last season, swept aside fellow German Daniel Farke’s team with their trademark attacking verve, taking a 3-0 lead inside 28 minutes.
Results 1-7