Let $L$ be a line bundle on a nodal curve $C$. Under which assumptions on $C$ and $L$ does there exist a "positive" Cartier divisor $D$ that "twists away" zeros of $L$ in the sense that it holds both $\mathcal{O}_C \subseteq \mathcal{O}_C(D)$ and $\mathcal{O}_C \subseteq L(D)$?
In the case of a smooth curve this is always the case as $L = \mathcal{O}_C(\sum_i m_i p_i - \sum_j n_jq_j)$ with $m_i, n_j \geq 0$ and we can choose $D= \sum_j n_jq_j$ as desired divisor. I am now interested if and if yes how I can generalize this technique if there is a twist at nodes.