The classical AT1 and AT2 phase-field models (bourdin et al. 2000, pham et al. 2011) were the first variational gradient damage models to successfully describe brittle fracture. Cohesive gradient damage models (cuvilliez et al. 2012), introduced later, allow the fracture energy Gc and tensile strength σc to be specified independently of the characteristic length lc, enabling more accurate and flexible modeling of crack nucleation and propagation in complex scenarios. Variational gradient damage models are advantageous because their numerical solution reduces to a constrained minimization problem over the discretized displacement "u" and damage field "α", which continuously describe cracks. Nevertheless, challenges remain due to the nonconvexity of the energy and the irreversibility of the nonlocal variable "α". In this talk, we leverage the Hybrid High Order method (HHO) (Di pietro et al. 2015) to solve the variational cohesive gradient damage model. Compared with the standard FEM method, HHO allows considering more general non conformal meshes and thus gives more flexibility in adaptative mesh refinement techniques. Regarding the irreversibility constraint, we follow the work of Siedel et al (seidel 2023), where a micromorphic approximation has been introduced. The micromorphic approach has the advantage of enforcing the irreversibility constraint at the integration point while still being variational and is able to approximate most damage gradient models with a unified implementation. It has however the disadvantage of introducing a penalisation coefficient which has no physical interpretation. The impact of this penalisation coeffcient is presented in (Nava Soto et al. 2026) where a new strategy based on the capacities of HHO method (static condensation) allow obtaining exact and robust treatment of irreversibility. This work validates this method on physical scenarii based on some of the test cases proposed in (kamarei et al. 2026), highlighting its ability to accurately predict crack nucleation and propagation in complex conditions, while assessing the impact of the stabilization constant on the HHO method to ensure robust and accurate results.