The Kolmogorov-Johnson-Mehl-Avrami model; which is a nucleation and growth Poissonian process in space; has been implemented by taking into account spatial correlation among nuclei. This is achieved through a detailed study of a system of distinguishable and correlated dots (nuclei). The probability that no dots be in a region of the space has been evaluated in terms of correlation functions. The central point of the paper is the application of the theory to describe nucleation and growth in two dimensions under constant nucleation rate, where correlation among nuclei depends upon the size of the nucleus. This is a typical case occurring in transformation governed by particle diffusion. We also propose a simple formula for describing the phase transition kinetics under this circumstance.