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We give a result concerning the correction to the Central Limit Theorem for a Random Walk on the lattice ${𝐙}^2$ which interacts with a random environment under a small randomness condition. Our main theorem close a gap which dates back to seminal works by Boldrighini, Minlos and Pellegrinotti, see [3], [8] and [9]. Asymptotic behaviour of the corrections to the average and the covariance matrix in dimension $\nu =1,2$ are also presented.