  For any $n\in \mathbb{Z}_{\geq 2}$, let $\mathfrak{m}_n$ be the subalgebra of
$\mathfrak{sp}_{2n}$ spanned by all long negative root vectors
$X_{-2\epsilon_i}$, $i=1,\dots,n$. An $\mathfrak{sp}_{2n}$-module $M$ is called
a Whittaker module with respect to the Whittaker pair
$(\mathfrak{sp}_{2n},\mathfrak{m}_n)$ if the action of $\mathfrak{m}_n$ on $M$
is locally finite, according to a definition of Batra and Mazorchuk. This kind
of modules are more general than the classical Whittaker modules defined by
Kostant. In this paper, we show that each non-singular block
$\mathcal{WH}_{\mathbf{a}}^{\mu}$ with finite dimensional Whittaker vector
subspaces is equivalent to a module category $\mathcal{W}^{\mathbf{a}}$ of the
even Weyl algebra $\mathcal{D}_n^{ev}$ which is semi-simple. As a corollary,
any simple module in the block
$\mathcal{WH}_{\mathbf{i}}^{-\frac{1}{2}\omega_n}$ for the fundamental weight
$\omega_n$ is equivalent to the Nilsson's module $N_{\mathbf{i}}$ up to an
automorphism of $\mathfrak{sp}_{2n}$. We also characterize all possible algebra
homomorphisms from $U(\mathfrak{sp}_{2n})$ to the Weyl algebra $\mathcal{D}_n$
under a natural condition.
