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However, the Harriers were allowed back in 15 minutes later thanks to a Damien Delaney mistake. His attempted clearance hit a player and John Williams capitalised to finish low past Paul Musselwhite.
Both sides had few opportunities, but Dean Bennett headed just wide of Musselwhite's goal after 33 minutes.
Four minutes before half time, City also went close to breaking the deadlock. Stuart Green took a free kick from the left flank, and Jason Price at the back post saw a header saved well by Stuart Brock.
But Brock was beaten two minutes later when Andy Dawson crossed and Danny Allsopp shot home to regain City's lead at half time.
City turned on the style in the second half, and 11 minutes in Andy Dawson scored a contender for goal of the season. He volleyed it from 25 yards after Stuart Elliott had laid it back, and Brock didn't move as it flew into the top corner.
City were cruising and Ben Burgess got his second of the match with 10 minutes to go. He controlled Stuart Green's corner at the near post and then put an overhead kick past Brock.
3 minutes later, Ryan France scored on his league debut after coming on as a 75th minute substitute. Andy Holt's cross was deflected into his path, and he poked the ball in to make it 5-1.
City closed the scoring at six with virtually the last kick of the match. Danny Williams gave away a free kick on the edge of the box, and a practiced set piece saw Stuart Green curl the ball in from a lay-off.
Despite the 6-1 win, City remain 2nd in the table as Swansea came from behind to beat Carlisle (click here for all of the day's Division 3 results).
City line-up (4-4-2): Musselwhite; Hinds, Whittle, Delaney, Dawson; Price (France), Green, Keates, Elliott (Holt); Allsopp (Forrester), Burgess.
Un-used subs: Fettis, Fry.