Many physical processes are controlled by the time that a random walker needs to reach a target. In many practical situations, such as reaction kinetics, this target is imperfect: multiple random encounters may be necessary to actually trigger a reaction. So far, most analytical approaches of imperfect reaction kinetics have been limited to Markovian (memoryless) stochastic processes. However, as soon as the random walker interacts with its environment, its motion becomes effectively non-Markovian. Here, we present a theory that provides the mean reaction time for a non-Markovian Gaussian random walker in a large confining volume in the presence of a spatially localized reaction rate or a gated target. Remarkably, in the weakly reactive regime, for strongly subdiffusive processes, our theory predicts that the deviation of the mean reaction time to the reaction controlled time displays a non-trivial scaling with the reactivity, which we identify analytically. This effect illustrates how the memory of past passages to the target influences the statistics of next-return times, to the difference of Markovian processes. The theory is developed in one and two dimensions and agrees with stochastic simulations. These results provide a refined understanding of how non-Markovian transport and local reactivity influence the kinetics of diffusion controlled reactions.