Core cutoff radius and energy cutoff radius in pseudo potential DFT
The reason pseudopotentials are use is because (at low pressures), core electrons do not participate in chemical bonding or reactions, and more over have high kinetic energies. If you expand and electronic wave function in a plane wave basis, high energy wave functions require high frequency basis functions, therefore in order to capture the wave function of a core electron accurately, you would need a high plane wave cutoff, which is computationally expensive.

To avoid this, the core electrons are replaced by a pseudopotential, which is by construction smooth. The core radius is to some extent a measure of the 'hardness' of the pseudopotential. A soft pseudopotential will generally have a larger core radius, and will be smooth. A hard pseudopotential will have a lower core radius, and in order to capture the rapidly varying (more jagged) potential near the core, will be less smooth, and therefore require higher frequencies in its Fourier transform to recover the correct electronic wave functions. Harder pseudopotential are necessary, for example, at high pressures, when the nuclei are squeezed together --- if the core parts of the pseudopotential overlap, then the calculation will fail catastrophically.