We predict hierarchies of kinetic regimes and crossover conditions for elementary A + A --> 0 and A + B --> 0 batch reactions from a nonclassical reaction-diffusion formalism that includes spatial fluctuations. This paper addresses the case of a spatially correlated initial distribution of reactants; random initial conditions have been discussed in another paper. The hierarchies of kinetic regimes are even richer than those found for random initial conditions, with the particular hierarchy determined by the tightness of the initial correlations. In most cases there is one regime (which we call a "burst") where the decay of the densities in the A + B reaction is even faster than in the A + A reaction. We confirm our predictions via detailed numerical simulations.