We present a version of the classical Moran model, in which mutations are taken into account; the possibility of mutations was introduced by Moran in his seminal paper, but it is more often overlooked in discussing the Moran model. For this model, fixation is prevented by mutation, and we have an ergodic Markov process; the equilibrium distribution for such a process was determined by Moran. The problems we consider in this paper are those of first hitting either one of the ``pure'' (uniform population) states, depending on the initial state; and that of first hitting times. The presence of mutations leads to a nonlinear dependence of the hitting probabilities on the initial state, and to a larger mean hitting time compared to the mutation-free process (in which case hitting corresponds to fixation of one of the alleles).