Strong (Inverse of) Residue Theorem

Indeed, this follows from Serre duality: Take your favorite local holomorphic coordinate $z$ centered at $p$, and use this as the transition function $g_{0,\infty}=z$ of the line bundle $L(p)$ with global holomorphic section $s_p$. Consider the section $$\omega\otimes s_{-np}$$ and take a $C^\infty$ cut-off away from $p$ as follows: Let $\varphi$ be a function with support in $\Delta$ which is constantly 1 near $p$. This gives a global section $\tilde\omega_n=\varphi \omega\otimes s_{n-p}\in \Gamma(C\setminus\{p\},K\otimes L(-np))$ which is meromorphic near $p$. Apply the $\bar\partial$ operator to obtain a smooth section $$\bar\partial\tilde\omega_n$$ of $\bar K KL(-np)$ with support in an annulus $A\subset\Delta,$ $p\notin A$. Consider a section $s\in H^0(C,L(np)).$ Then $$\int_C \bar\partial(\tilde\omega_n) s=\int_C(\bar\partial \varphi) \omega s_{-np} s=\int_{\partial A}\varphi \omega s_{-np} s=-\int_\gamma \omega s_{-np}s=-2\pi i Res_p(f\omega),$$ where $\gamma$ is a small curve around $p$ along which $\varphi$ is 1, and $f$ is the meromorphic function $f=s_{-np}s.$ Therefore, by your assumption, the pairing of $\bar\partial\tilde\omega_n$ with any holomorphic section in $L(np)$ vanishes, and Serre duality yields a smooth section $s$ of $KL(-np)$ such that $$\bar\partial\tilde \omega_n=\bar\partial s_n,$$ hence $$\tilde{\omega}_n-s_n$$ is a global meromorphic section of $KL(-np),$ or equivalently, a meromorphic 1-form $\omega_n$ with prescribed behavior up to order $n$ around $p.$ For $n$ big enough ($\geq 2g-2$), all $\omega_n$ are the same (e.g., by the easy part of the Serre duality) and hence coincide with $\omega $ on $\Delta\setminus\{p\}$.