Bonded brickwork used for loadbearing walls is widely found in heritage structures worldwide. The evaluation of bonded masonry structures and their strengthening strategies against dynamic actions require appropriate understanding under cyclic loading. Subsequently, a simplified 3D microscale numerical model is developed in this paper to analyse bonded brickwork under cyclic compression. A plasticity-based damage constitutive model to represent damage in masonry bricks under cyclic compression loading was employed, and zero-thickness interfaces were considered with non-linear damage properties to simulate the mechanical behaviour of masonry. A threshold strain level was used to enact the element deletion technique for initiating brittle crack opening in the masonry units. The developed model was validated against the experimental results published by the authors in the past. The models were able to accurately predict the experimental results with an error limit of 10% maximum. Mainly, two types of strengthening materials, possessing (1) high energy absorption characteristics (auxetic fabric) and (2) high strength properties (carbon fibre reinforced polymer composites/CFRP) were employed for damage mitigation under cyclic compression. Results show that the CFRP-strengthened masonry failure was mainly attributed to de-bonding of the CFRP and crushing under compression. However, the auxetic strengthening is shown to significantly minimise the de-bonding phenomenon. Enhanced energy dissipation characteristics with relatively higher ductility (up to ~50%) and reduced damages on the bonded brickwork were observed as compared to the CFRP-strengthened brickwork under cyclic compression loading. Additionally, the auxetic fabric application also increased the compressive resistance of brickwork by 38–60% under monotonic loading, which is comparably higher than with the CFRP strengthening technique.