The authors prove the existence of a periodic solution to a scalar second order differential equation of the type \[ u''=h(t)g(u), \] where \(g\in C^1({\mathbb R}_+;{\mathbb R}_+)\) is a nondecreasing function with a singularity at zero, satisfying \[ \int_0^1g(s)\,ds=+\infty, \] and \(h\in L^1_{\text{loc}}({\mathbb R})\) is a sign-changing \(T\)-periodic function, with negative mean. As a typical model, they have in mind a generalized Emden-Fowler equation like \[ u''=\frac{h(t)}{u^\mu}, \] where \(\mu\geq1\). Using topological degree arguments, they are able to deal with several different cases, where the weight function \(h(t)\) interacts with the function \(g\) (in particular, with the exponent \(\mu\)). The analysis of such problems is rather delicate, and the interesting results of this paper should be further pursued, in order to reach a complete description of more general situations.