In this contribution, we extend physics-augmented neural network-based constitutive models for hyperelasticity to softening in terms of the Mullins effect, which enables the material modeling of soft rubbers, polymers and metamaterials with stress-softening or damage in cyclic loading. The pseudo-elastic approach is based on the formulation of a hyperelastic strain energy function using a physics-augmented neural network, which is expressed in terms of isotropic strain invariants and ensures objectivity, material symmetry, energy and stress normalization, and volumetric growth conditions [1,2]. To model softening, this strain energy term is scaled with a damage factor. As in the Ogden-Roxburgh model [3], the damage factor depends on the current and the maximum experienced strain energy, but here it is computed using a combination of a standard feed forward and a monotonic neural network. With this physics-augmented architecture, a physically sound behavior of the damage factor and the thermo-dynamic consistency of the model are ensured. This novel model formulation is trained on cyclic uniaxial, biaxial, and shear loading scenarios with data generated from an analytical model for two sulphur EPDM material compounds, which is conceptually different as it uses an amplification approach instead of the damage factor [4]. While the Ogden-Roxburgh model cannot accurately capture this behavior, very accurate predictions can be obtained with the proposed model due to the flexibility of the neural networks and due to the physics-augmentation also reasonable extrapolation results can be obtained.