This short paper presents a general approach for computing robust Wasserstein barycenters (Agueh et al. 2011), (Turner et al. 2014),(Vidal et al. 2020) of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this procedure only works for the transportation cost related to the $q$q-Wasserstein distance $W_{q}$Wq when $q=2$q=2. We adapt an alternative fixed-point method (Tanguy et al. 2025) to compute a barycenter diagram for generic transportation costs ($q > 1$q>1), in particular those robust to outliers, $q \in (1,2)$q∈(1,2). We show the utility of our work in two applications: (i) the clustering of persistence diagrams on their metric space and (ii) the dictionary encoding of persistence diagrams (Sisouk et al. 2024). In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework.