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Forward Bakkor, 25, is no stranger to Wincham Park having made 40 appearances and scoring nine goals for Albion between 2013 and 2014.
He returns to the club after spending the last two seasons with fellow Northern Premier League Premier Division outfit Stalybridge Celtic, with whom he won their Player of the Season award in 2018-19.
Comfortable in either wide position as well as down the middle, Bakkor has also played for Colwyn Bay, Clitheroe, Marine, Warrington Town, Curzon Ashton and Skelmersdale United.
Midfielder Gregory arrives at Albion after playing the 2019-20 campaign with Northern Premier League North West Division side Prescot Cables.
The 21-year-old was a regular in the heart of the Cables midfield and began his career coming through the Academy at Wigan Athletic before spells with Chorley and Clitheroe.
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