Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in C^\infty_c(M \setminus \{q\})$, that is, $\varphi$ is smooth and compactly supported away from $q$, and $C$ is independent of $\varphi$ and $u$. If we know that $u \in H^1 \cap L^p$, can we conclude from the above that $$\Vert u\Vert_{L^p} \leq C\Vert u\Vert_{H^1}?$$

Edit: It seems that it suffices to claim that there exists a sequence $\varphi_k$ such that $\varphi_k u \to u$ in $L^p$-norm and $\varphi_k u \to u$ in $H^1$-norm. But I cannot justify the existence of such a sequence.