The Kontsevich--Zorich cocycle over Veech--McMullen family of symmetric translation surfaces

We describe the Kontsevich–Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich–Zorich monodromies of $SU(p,q)$ type are realized by appropriate covering constructions.