  We study the stack $\mathcal{H}_{r,g,n}$ of $n$-pointed smooth cyclic covers
of degree $r$ between smooth curves of genus $g$ and the projective line. We
give two presentations of an open substack of $\mathcal{H}_{r,g,n}$ as a
quotient stack, and we study its complement. Using this, we compute the
integral Picard group of $\mathcal{H}_{r,g,n}$. Moreover, we obtain a very
explicit description of the generators of the Picard group, which have evident
geometric meaning. As a corollary of the computation, we get the integral
Picard group of the stack $\mathcal{H}_{g,n}$ of $n$-pointed hyperelliptic
curves of genus $g$. Finally, taking $g=2$ and recalling that
$\mathcal{H}_{2,n}=\mathcal{M}_{2,n}$, we obtain
$\mathrm{Pic}(\mathcal{M}_{2,n})$.
