We study skew-products of the form $(x,\omega)\mapsto (Tx, \omega+\phi(x))$ , where $T$ is a nonuniformly expanding map on a space $X$ , preserving a (possibly singular) probability measure $\tilde\mu$ , and $\phi:X\to \mathbb{S}^1$ is a $C^1$ function. Under mild assumptions on $\tilde\mu$ and $\phi$ , we prove that such a map is exponentially mixing and satisfies both the central limit and local limit theorems. These results apply to a random walk related to the Farey sequence, thereby answering a question of Guivarc'h and Raugi [GR, Section 5.3]