By way of the projected phase space (PPS), we investigate the relation between galaxy properties and cluster environment in a subsample of groups from the Yang catalogue. The sample is split according to the Gaussianity of the velocity distribution in the group into Gaussian (G) and non-Gaussian (NG). Our sample is limited to massive clusters with $\rm \mathit{ M}_{200} \ge 10^{14}\, M_{\odot }$ and $\rm 0.03\le \mathit{ z} \le 0.1$ . NG clusters are more massive, less concentrated and have an excess of faint (F) galaxies compared to G clusters. NG clusters show mixed distributions of galaxy properties in the PPS compared to the G case. Using the relation between infall time and locus on the PPS, we find that, on average, NG clusters accreted ${\sim}\rm 10^{11}\, M_{\odot }$ more stellar mass in the last ∼5 Gyr than G clusters. The relation between galaxy properties and infall time is significantly different for galaxies in G and NG systems. The more mixed distribution in the PPS of NG clusters translates into shallower relations with infall time. F galaxies whose first crossing of the cluster virial radius happened 2-4 Gyr ago in NG clusters are older and more metal-rich than in G systems. All these results suggest that NG clusters experience a higher accretion of pre-processed galaxies, which characterizes G and NG clusters as different environments to study galaxy evolution.