Extension of regular function

Solution 1:

Since the problem is local around $p$, you can assume that $Y=Spec(A)$ where $A$ is a noetherian domain (quasi-projectiveness is irrelevant).

Clearly, every point $\mathfrak q \in Spec(A)$ of height one is distinct from $p$ (since $\mathfrak q$ it corresponds to a subvariety of codimension $1$). So every function $f$ defined on $Spec(A)\setminus \lbrace p\rbrace $ is defined at $\mathfrak q $.
You can then conclude that $f\in A$, that is $f$ extends regularly through $p$, thanks to the formula valid for a noetherian normal domain (Matsumura, Commutative ring theory, Theorem 11.5, page 81)

$$ A=\bigcap_{ht(\mathfrak q)=1} A_ \mathfrak q $$

A general result in this vein is that if $X$ is a locally noetherian normal integral scheme and $Y\subset X$ a closed subset of codimension $\geq 2$, the restriction morphism $\mathcal O_X(X)\to O_X(X\setminus F)$ is bijective.