Summary: A proper total weighting of a graph \(G\) is a mapping \(\phi\) which assigns to each vertex and each edge of \(G\) a real number as its weight so that for any edge \(u v\) of \(G\), \(\sum_{e \in E(v)} \phi(e) + \phi(v) \neq \sum_{e \in E(u)} \phi(e) + \phi(u)\). A \((k, k^\prime)\)-list assignment of \(G\) is a mapping \(L\) which assigns to each vertex \(v\) a set \(L(v)\) of \(k\) permissible weights and to each edge \(e\) a set \(L(e)\) of \(k^\prime\) permissible weights. An \(L\)-total weighting is a total weighting \(\phi\) with \(\phi(z) \in L(z)\) for each \(z \in V(G) \cup E(G)\). A graph \(G\) is called \((k, k^\prime)\)-choosable if for every \((k, k^\prime)\)-list assignment \(L\) of \(G\), there exists a proper \(L\)-total weighting. It was proved in [Y. Tang and X. Zhu, Discrete Math. 340, No. 8, 2033–2042 (2017; Zbl 1362.05057)] that if \(p \in \{5, 7, 11 \}\), a graph \(G\) without isolated edges and with \(\operatorname{mad}(G) \leq p - 1\) is \((1, p)\)-choosable. In this paper, we strengthen this result by showing that for any prime \(p\), a graph \(G\) without isolated edges and with \(\operatorname{mad}(G) \leq p - 1\) is \((1, p)\)-choosable.