  We study fluctuations in the number of points of $\ell$-cyclic covers of the
projective line over the finite field $\mathbb{F}_q$ when $q \equiv 1 \mod
\ell$ is fixed and the genus tends to infinity. The distribution is given as a
sum of $q+1$ i.i.d. random variables. This was settled for hyperelliptic curves
by Kurlberg and Rudnick, while statistics were obtained for certain components
of the moduli space of $\ell$-cyclic covers by Bucur, David, Feigon and
Lal\'{i}n. In this paper, we obtain statistics for the distribution of the
number of points as the covers vary over the full moduli space of $\ell$-cyclic
covers of genus $g$. This is achieved by relating $\ell$-covers to cyclic
function field extensions, and counting such extensions with prescribed
ramification and splitting conditions at a finite number of primes.
