  Let $R$ be a ring and $n$, $k$ two non-negative integers. In this paper, we
introduce the concepts of $n$-weak injective and $n$-weak flat modules and via
the notion of special super finitely presented modules, we obtain some
characterizations of these modules. We also investigate two classes of modules
with richer contents, namely $\mathcal{WI}_k^n(R)$ and
$\mathcal{WF}_k^n(R^{op})$ which are larger than that of modules with weak
injective and weak flat dimensions less than or equal to $k$. Then on any
arbitrary ring, we study the existence of $\mathcal{WI}_k^n(R)$ and
$\mathcal{WF}_k^n(R^{op})$ covers and preenvelopes
