The 28-year-old was former manager Gary Brabin’s first signing of the summer after his release from Northampton Town.
He started eight out of the first nine league matches, before being sent off at home to Lincoln on September 17.
Injury and the outstanding form of Janos Kovacs and then restricted him to just three more appearances in cup competitions for the remainder of the campaign.
Beckwith said: “It’s been really frustrating, so I’ll have a word with the manager, see what he wants and take it from there.
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“If the manager wants me next season, I’ll work as hard as I have been working. I’ll keep fit over the summer, I always have done and then hopefully come back, start well pre-season and go from there.”
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