Klukowski, 21, scored twice for the League One side in Saturday's 3-1 victory at Workington, but Abbott will not be offering him a permanent contract.
Abbott told the club's official website: "Yan made this a very difficult decision for me to make because he worked really hard while he was with us.
"He is a strong player but, as with every trialist we look at, we have to be certain that they are better than what we already have.
He got his goals, but we're just not sure that he will make it to the next level.
"He can go away with his head high though, and we wish him all the best for the future."