EDIT : Olivier Wittenberg pointed out to me that a positive answer to the question follows from Theorem 1 of [Altman-Kleiman, Bertini theorems for hypersurface sections containing a subscheme]. I keep below my previous answer (whose argument is different, but more complicated).

The answer to your question is positive. Let $X$ be an integral projective variety of dimension $n\geq 2$ (over an algebraically closed field) and $V\subset X$ a subvariety of codimension $\geq 2$. Let $A$ be ample on $X$ : up to replacing $A$ by a multiple, we may assume that $X\subset\mathbb{P}^N$ and $A=\mathcal{O}_X(1)$.

Let $\mathcal{H}_e$ be the projective space of degree $e$ hypersurfaces in $\mathbb{P}^N$,
$\mathcal{F}_e^{igr}(X)\subset\mathcal{H}_e$ the subset consisting of hypersurfaces whose intersections with $X$ are not irreducible generically reduced of codimension $1$ in $X$, and
$\mathcal{G}_e(V)\subset\mathcal{H}_e$ the subset consisting of hypersurfaces containing $V$.

The exact sequence $0\to\mathcal{I}_V(e)\to\mathcal{O}_{\mathbb{P}^N}(e)\to\mathcal{O}_V(e)\to 0$ shows that, when $
e\gg 0$, the codimension of $\mathcal{G}_e(V)$ in $\mathcal{H}_e$ is a polynomial of degree $\leq n-2$ in $e$ (the Hilbert polynomial of $V$). On the other hand, Théorème 0.4 of arXiv:0911.1118 shows that, when $e\geq 2$, the codimension of $\mathcal{F}_e^{igr}(X)$ in $\mathcal{H}_e$ is $\geq \binom{e+n-1}{n-1}-n$, that is at least a polynomial of degree $n-1$ in $e$. As a consequence, if $e \gg 0$, $\mathcal{G}_e(V)$ is not included in $\mathcal{F}_e^{igr}(X)$. A hypersurface in $\mathcal{G}_e(V)$ but not in $\mathcal{F}_e^{igr}(X)$ induces on $X$ the divisor you are looking for.

Note that if $X$ is moreover $S_2$ (for instance, normal), then this divisor is itself integral.