Cutting and groove-packing of hard-packed snow in tyre–snow interaction are governed by strong degradation and partial recovery of strength under large deformation. Elastoplastic models with algebraic damage–healing formulations are often used to describe such behaviour in different materials. These formulations reproduce basic softening and healing trends, but they contain a structural limitation: once the healing variable approaches its upper bound, further damage under renewed shear cannot be represented consistently. In this study, for the first time for hard-packed snow, a rate-type scalar damage–healing variable is introduced within a finite-strain cohesive Modified Cam-Clay (MCC) framework. Damage and healing rates are driven by measures of deviatoric and volumetric plastic deformation, so that the internal variable can increase and decrease repeatedly along complex loading paths. An implicit gradient enhancement is adopted to regularise softening and to introduce an internal length scale. The model is implemented in (a) a standard finite element formulation and (b) an explicit B-spline Material Point Method (MPM). The behaviour is first examined by four material tests: (i) Oedometer compression, (ii) triaxial compression, (iii) uniaxial compression, and (iv) direct shear. In these tests, realistic stress–strain responses are obtained and localisation is rendered objective with respect to mesh and particle resolution. Three large-deformation cutting configurations are then simulated with MPM: ( i ) a blade-cutting test, ( ii ) a forward snow-cavity (SC) test, and ( iii ) a backward SC test, representing milling, groove filling, and snow-snow friction. Four constitutive variants are compared: (1) purely elastoplastic MCC, (2) MCC with damage only, (3) MCC with algebraic damage–healing, and (4) the proposed rate-type damage–healing law. The results show that the algebraic formulation can reproduce basic degradation and recovery trends, but fails to represent re-damage after strong healing in reversed or repeated shear loading. In contrast, the proposed rate-type gradient-enhanced model reproduces the measured force evolution and the experimentally observed deformation patterns in blade-cutting and SC tests, including the transition from groove filling to snow-snow sliding. The proposed framework therefore provides a consistent constitutive basis for hard-packed snow in process-scale tyre applications.