If I understand the question correctly, what you are after is an effective Hamiltonian action of $\mathbb{T}^m$ on a (closed) symplectic manifold $(M,\omega)$ such that there exists a point $p \in M$ with finite stabilizer. If this is the case, there are plenty of examples. Consider a closed symplectic toric manifold $\mu: (M^4,\omega) \to \mathfrak{t}^*$ with Delzant (= moment) polygon $\Delta = \mu(M)$ with the property that the primitive integral tangent vector to the edge $e$ of $\Delta$ is of the form $(k,a) \in \mathbb{Z}^2$ for $|k| > 1$. Writing $\mathbb{T}^2 = S^1 \times S^1$, consider the restriction of this action to the first $S^1$, thus obtaining what is known as a Hamiltonian $S^1$-space $S^1 \curvearrowright (M^4,\omega)$ (these have been classified by Karshon in this paper). The moment map of this action is the composition $\mathrm{pr}_1 \circ \mu : M \to \mathrm{Lie}(S^1)^* \cong \mathbb{R}$, where $\mathrm{pr}_1 : \mathrm{t}^* \to \mathrm{Lie}(S^1)^*$ is the projection induced by the inclusion $S^1 \hookrightarrow \mathbb{T}^2$. Any point $p \in \mu^{-1}(\mathring{e})$ has finite stabilizer with respect to this $S^1$-action; in fact, the stabilizer is precisely the cyclic group of order $k$!

There are examples of Hamiltonian $S^1$-spaces which do not arise as above (*i.e.* they cannot be extended to a toric action) which also have finite stabilizers. If you take a look at Karshon's classification, these are precisely the points lying on what she calls $\mathbb{Z}_k$-spheres (except for the "north" and "south" poles of the spheres which are fixed points).

In all of the above examples, the set of points with finite stabilizer is "small", *i.e.* it has empty interior. This is always the case for *effective* Hamiltonian actions by a torus $\mathbb{T}^m$ on a connected symplectic manifold $(M,\omega)$. The fact that $\mathbb{T}^m$ is abelian is crucial in the above statement. If you are interested in the case of this set of points with finite stabiliser being "large", say being open and dense, then a good place to look at is *multiplicity-free Hamiltonian spaces*, which are the non-abelian analogue of closed symplectic toric manifolds. A good reference is this paper by Knop.