  As a consequence of the main result of this paper efficient conditions
guaranteeing the existence of a $T-$periodic solution to the second order
differential equation \begin{equation*} u"=\frac{h(t)}{u^{\lambda}}
\end{equation*} are established. Here, $h\in L(\mathbb{R}/T\mathbb{Z})$ is a
piecewise-constant sign-changing function where the non-linear term presents a
weak singularity at 0 (i.e. $\lambda\in (0,1)$).
