Let $X \to P^2$ be a $p$-cyclic cover branched over a smooth, connected curve $C$ of degree divisible by $p$, defined over a separably closed field of prime-to-$p$ characteristic. We define a homomorphism $(Pic C/Z L)[p] \to Br X[p]$, which factors through $Br k(P^2)$. In addition, the image contains all Brauer classes on $X$ that are fixed by $Aut(X/P^2)$. If $p = 2$, we give a geometric construction, which works over any field of characteristic not 2, that uses Clifford algebras arising from symmetric resolutions of line bundles on $C$ to yield Azumaya representatives for the 2-torsion Brauer classes on $X$. We show that, when $p=2$, both constructions give the same result. This generalizes work of van Geemen for degree 2 K3 surfaces with Picard rank 1. This is joint work with Colin Ingalls, Andrew Obus, and Ekin Ozman.