Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $mathfrak{sl}_2$ using singular blocks of category $mathcal{O}$ for $mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $mathfrak{sl}_n$ over a field $textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $mathfrak{sl}_k$-action, following Sussans approach, by considering more singular blocks of modular representations of $mathfrak{sl}_n$. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of $mathfrak{sl}_n$.
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