The `core-cusp' problem has long been considered a key small scale challenge to the standard LCDM paradigm. Halos in dark matter only simulations exhibit `cuspy' (NFW-like) density profiles, where density continuously increases towards the centre. However, many observations of galaxies (particularly in the dwarf regime) indicate that their dark matter density profiles deviate strongly from this prediction, with much flatter central regions (`cores'). Here, we use NewHorizon (NH), a hydrodynamical cosmological simulation which adopts the standard LCDM cosmology, to investigate core formation and the core-cusp problem, using a statistically significant number of galaxies in a cosmological volume. In NH, halos that contain galaxies in the upper M* > 10^10.2 MSun) and lower (M* < 10^8 MSun) ends of the stellar mass distribution contain cusps. However, halos that contain galaxies with intermediate (10^8 MSun < M* < 10^10.2 MSun) stellar masses are typically cored. These halos typically have DM masses between 10^10.2 MSun and 10^11.5 MSun. Cores are created via supernova-driven gas removal from the central regions of halos, which alters the central gravitational potential, inducing dark matter to migrate to larger radii. While all massive (M* > 10^9.5 MSun) galaxies undergo a cored-phase, in some cases cores can be removed and cusps reformed. This happens if a galaxy undergoes sustained star formation at high redshift, which results in stars (which, unlike the gas, cannot be removed by baryonic feedback) dominating the central gravitational potential. After cosmic star formation peaks, the number of cores and the mass of the halos they are formed in remain constant throughout the simulation, indicating that cores are being routinely formed over cosmic time after a threshold halo mass is reached. The existence of cores is, therefore, not in tension with the standard paradigm.