Let $Y$ be a normal surface and let $X$ be a closed subscheme of codimension 2, i.e., $X$ is a finite set of closed points.
Let $D$ be a Weil divisor on $Y$.
Question. Does there exist a Weil divisor $E$ on $Y$ which is linearly equivalent to $D$ and does not go through $X$? (Edit: I do not assume $E$ to be effective.)
Of course, when $D$ does not go through $X$ the answer is yes.
When I say that $E$ does not go through $X$, I mean that the support of $E$ and $X$ are disjoint.
I'm interested in this question in the most general set-up known. For example, $Y$ is an integral noetherian separated excellent normal 2-dimensional scheme. If you prefer sticking to algebraic surfaces, I would be glad to hear about what's possible in that case too.
The motivation for this question comes from an article by Mumford in which he defines an intersection pairing on a normal surface. In this case $X$ is the singular locus of $Y$.