Postirradiation dissociation of molecular clusters has been mainly studied assuming energy redistribution in the entire cluster prior to the dissociation. Here, the evaporation of water molecules from out-of-equilibrium pyridinium-water cluster ions was investigated using the recently developed correlated ion and neutral time-of-flight (COINTOF) mass spectrometry technique in combination with a velocity-map imaging (VMI) device. This special setup enables the measurement of velocity distributions of the evaporated molecules upon high-velocity collisions with an argon atom. The distributions measured for pyridinium-water cluster ions are found to have two distinct components. Besides a low-velocity contribution, which corresponds to the statistical evaporation of water molecules after nearly complete redistribution of the excitation energy within the clusters, a high-velocity contribution is also found in which the molecules are evaporated before the energy redistribution is complete. These two different evaporation modes were previously observed and described for protonated water cluster ions. However, unlike in the case of pure water clusters, the low-velocity part of the distributions for pyridinium-doped water clusters is itself composed of two distinct Maxwell-Boltzmann distributions, indicating that evaporated molecules originate in this case from out-of-equilibrium processes. Statistical molecular dynamics simulations were performed to (i) understand the effects caused in the ensuing evaporation process by the various excitation modes at different initial cluster constituents and to (ii) simulate the distributions resulting from sequential evaporations. The presence of a hydrophobic impurity in water clusters is shown to impact water molecule evaporation due to the energy storage in the internal degrees of freedom of the impurity.