We formulate physics-augmented neural network (PANN)-based constitutive models for Mullins softening material behavior. This enables the constitutive modeling of soft rubbers, polymers, composites and metamaterials with stress-softening or damage in cyclic loading. The pseudo-elastic approach [1] is based on an isotropic hyperelastic PANN model [2] which is complemented by a damage function to model softening. The damage function is based on feed-forward neural networks, for which we investigate the influence of different monotonicity and convexity constraints. We furthermore consider parametric dependencies of the material behavior. With this PANN architecture, a physically sound behavior of the damage factor and the thermodynamic consistency of the model are ensured. We apply the PANN model to multiaxial, cyclic experimental data of an EPDM material synthesised with different amounts of sulphur contents [3]. The PANN model shows an excellent performance both in representing the ground truth material behavior and in nonlinear finite element analysis. [1] Ogden, R. W. & Roxburgh, D. G. (1988). A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society of London. Series A, 455, 2861-2877. [2] Linden, L., Klein, D. K., Kalina, K. A., Brummund, J., Weeger, O., & Kästner, M. (2023). Neural networks meet hyperelasticity: A guide to enforcing physics. Journal of the Mechanics and Physics of Solids, 179, 105363. [3] Plagge, J. & Klüppel, M. (2017). A physically based model of stress softening and hysteresis of filled rubber including rate- and temperature dependency. International Journal of Plasticity, 89.