In this paper, the periodic problem is studied for the following Liénard equation with a singularity of indefinite type: \(\begin{aligned} x''+f(x)x'-h(t)x^\delta -\frac{a(t)}{x^\mu }=s, \end{aligned}\) where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is continuous, \(a,h:\mathbb {R}/T\mathbb {Z}\rightarrow \mathbb {R}\) are continuous with h is positive and \(\bar{a}>0\) , \(s,\delta \) and \(\mu \) are constants with \(\delta >0\) and \(\mu >0\) . We consider the situation where the regular restoring force term \( h(t)x^\delta (t)\) is allowed to satisfy superlinear growth condition. By means of lower and upper function theory, as well as Leray–Schauder degree theory, some Ambrosetti–Prodi type results are obtained. Moreover, the asymptotic behavior of periodic solutions is investigated with respect to the parameter s.
