The victory has moved Coventry to within a point of sixth-placed United in the table
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Clarke put the visitors ahead in the 14th minute when he had the simple task of tapping in from six yards following Stephen Elliott's cross.
United made their second-half pressure pay thanks to David Kitson's header from a corner on 71 minutes.
However, Coventry snatched the win with three minutes left to play as Clarke converted the rebound after Franck Moussa's initial effort had been parried.