The evolution of a particle undergoing a continuous-time random walk in the presence of randomly placed imperfectly absorbing traps is studied. At long times, the spatial probability distribution becomes strongly localized in a sequence of trap-free regions. The subsequent intermittent transfer of the survival probability from small trap-free regions to larger trap-free regions is described as a time-directed variable range hopping among localized eigenstates in the Lifshitz tail. An asymptotic expression for the configurational average of the spatial distribution of surviving particles is obtained based on this description. The distribution is an exponential function of distance which expands superdiffusively, with the mean-square displacement increasing with time as t(2)/ln((2D+4)/D)(t) in D dimensions.