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Let be a ring graded by a group and an integer. We introduce the notion of -FCP-gr-projective -modules and by using of special finitely copresented graded modules, we investigate that (1) there exist some equivalent characterizations of -FCP-gr-projective modules and graded right modules of -FCP-gr-projective dimension at most over -gr-cocoherent rings, (2) is right -gr-cocoherent if and only if for every short exact sequence of graded right -modules, where and are -FCP-gr-projective, it follows that is -FCP-gr-projective if and only if (-, -) is a hereditary cotorsion theory, where - denotes the class of -FCP-gr-projective right modules. Then we examine some of the conditions equivalent to that each right -module is -FCP-gr-projective.