We consider a partially hyperbolic set K on a Riemannian manifold M whose tangent space splits as T_K M = E^{cu} ⊕ E^s, for which the center-unstable direction E^{cu} expands non-uniformly on some local unstable disk. We prove that the (stretched) expo- nential decay of recurrence time can be deduced in terms of the (stretched) exponential decay of the time that typical points need to achieve some uniform expanding behavior in the center-unstable direction. Here we give a local Gibbs-Markov-Young structure which plays a preponderant role. As an application of the main result we obtain (stretched) exponential decay of correlations and exponentially large deviation for the system.