n-Super finitely copresented and n-weak projective modules

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