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Let be a  ring and a non-negative integer. In this paper,  we first introduce the concept of -super finitely copresented  -modules and via these modules, we give a concept of -weak projective modules and investigate some characterizations of these modules over any arbitrary ring. For example, we obtain that () is a perfect hereditary cotorsion theory and for any , there exists an -weak projective cover with the unique mapping property if and only if every -module is  -weak projective.