  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 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 the central 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.
