  We extend the single-site coherent potential approximation (CPA) to include
the effects of non-local disorder correlations (alloy short-range order) on the
electronic structure of random alloy systems. This is achieved by mapping the
original Anderson disorder problem to that of a selfconsistently embedded
cluster. This cluster problem is then solved using the equations of motion
technique. The CPA is recovered for cluster size $N_{c}=1$, and the disorder
averaged density-of-states (DOS) is always positive definite. Various new
features, compared to those observed in CPA, and related to repeated scattering
on pairs of sites, reflecting the effect of SRO are clearly visible in the DOS.
It is explicitly shown that the cluster-CPA method always yields
positive-definite DOS. Anderson localization effects have been investigated
within this approach. In general, we find that Anderson localization sets in
before band splitting occurs, and that increasing partial order drives a
continuous transition from an Anderson insulator to an incoherent metal.
