Exponential mixing for singular skew-products

We study skew-products of the form $(x,u) \mapsto (fx, u + \varphi(x))$ where $f$ is a non-uniformly expanding map on a manifold $X$ and $\varphi: X \to \mathbb{S}^1$ is piecewise $\mathcal{C}^1$. If the systems satisfies mild assumptions (in particular singular behaviour of $\varphi$ is permitted) then we prove that the map mixes exponentially with respect to the unique SRB measure. This extends previous results by allowing singular behaviour in the fibre map.