  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.
