The ex-Leeds and Liverpool forward, who was only appointed at County on August 31 following an impressive stint at Crawley, was axed on Tuesday with his side sitting just outside the League Two relegation zone.
The 40 year-old, who had signed a three-year contract to replace Kevin Nolan at Meadow Lane, won just three of his 14 matches in charge and departs the club along with assistant manager Warren Feeney and first-team analyst Paul Murphy.
County chairman Alan Hardy said the board believed "quick action was essential".
"Harry's passion and commitment as a coach is unquestionable. I enjoyed seeing first-hand his love for the job and it gives me absolutely no pleasure to take that away from him," he said in a statement released on the club's website.
"Unfortunately, however, things quite simply were not working out for him here at Meadow Lane.
"Results have not been good enough and performance levels are a continuing concern. We see no reason to continue with something we don't think can work.
"Harry leaves with our best wishes for the future and our search for a new manager begins immediately.
"We hope to make an appointment before Saturday's Sky Bet League Two match at home to Cheltenham Town."
Steve Chettle will oversee County's Checkatrade Trophy clash at Doncaster on Wednesday night.
