  We consider a perturbation of a central force problem of the form
\begin{equation*} \ddot x = V'(|x|) \frac{x}{|x|} + \varepsilon \,\nabla_x
U(t,x), \quad x \in \mathbb{R}^{2} \setminus \{0\}, \end{equation*} where
$\varepsilon \in \mathbb{R}$ is a small parameter, $V\colon (0,+\infty) \to
\mathbb{R}$ and $U\colon \mathbb{R} \times (\mathbb{R}^{2} \setminus \{0\}) \to
\mathbb{R}$ are smooth functions, and $U$ is $\tau$-periodic in the first
variable. Based on the introduction of suitable time-maps (the radial period
and the apsidal angle) for the unperturbed problem ($\varepsilon=0$) and of an
associated non-degeneracy condition, we apply an higher-dimensional version of
the Poincar\'{e}-Birkhoff fixed point theorem to prove the existence of
non-circular $\tau$-periodic solutions bifurcating from invariant tori at
$\varepsilon=0$. We then prove that this non-degeneracy condition is satisfied
for some concrete examples of physical interest (including the homogeneous
potential $V(r)=\kappa/r^{\alpha}$ for
$\alpha\in(-\infty,2)\setminus\{-2,0,1\}$). Finally, an application is given to
a restricted $3$-body problem with a non-Newtonian interaction.
