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Spink only took charge in the summer, but departs with the club bottom of the league table with five defeats from five games.
They also crashed out of the FA Cup at home to Tadcaster Albion.
His final game in charge was the 3-1 home defeat to Worksop Town on Saturday.
A club statement said: "The club would like to thank Craig for his hard work and dedication during this period and we wish him and his family all the very best for the future.
"Sporting Director Spencer Fearn will take temporary charge of the team, whilst the club recruits a new Manager."
Applications for the post are to be sent to firstname.lastname@example.org. Deadline is Saturday 14th September 2019 at 5pm.
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