Let $C$ be a reduced affine Noetherian scheme of pure dimension 1 (all its irreducible components have dimension 1) and $p \in C$ a regular closed point. Suppose we have a morphism $C \backslash p \to \mathbb{P}^n$ that lies in a closed subscheme $Y \subseteq \mathbb{P}^n$. If this morphism extends to $C \to \mathbb{P}^n$, is it true that this extension must factor through $Y$?
Here's my proof. Since $p$ is a closed point, $p \in C' \subset C$ some irreducible component of $C$, necessarily of dimension 1. Note $C'$ is an irreducible Noetherian topological space of dimension 1 and $\pi^{-1}(Y) \cap C'$ is a closed subset of $C'$ containing $C' \backslash p$. However, any proper closed subset of an irreducible Noetherian topological space of dimension 1 must have dimension 0 and thus be finite. But $C'$ is infinite as it has dimension 1 so $\pi^{-1}(Y) \cap C' = C'$ and hence $\pi^{-1}(Y) \ni p$. Now, since $C$ is reduced the schematic image is the closure of the image, but the image is contained in $Y$, and $Y$ is closed in $\mathbb{P}^n$ and we conclude the schematic image of $C$ is contained in $Y$.
Yes, this is fine - it may be quicker to note that morphisms of schemes are continuous and that $f(\overline{X})\subset \overline{f(X)}$ for a continuous map of topological spaces $f:S\to T$ and $X\subset S$.
For other extension problems about morphisms of schemes, you may wish to look up the notions of separated and proper.