The Material Point Method (MPM) is an established approach for modelling solid mechanics problems involving large deformation and non-linear, history dependent, behaviour. This is achieved by relaxing the coupling of the deformation of the physical material with the underlying computational mesh. The physical body is represented by a collection of material points, which carry information about the material, such as mass, volume, deformation, stress, etc., this information is mapping to a background (finite element) grid where the governing equations are solved. Once equilibrium has been obtained, the material point positions and states (deformation, stress, etc.) are updated, whereas the deformed grid is discarded and a new grid introduced for the subsequent step. This avoids mesh distortion/entanglement, which can take place in finite element methods. However, decoupling the deformation of the material from the computational mesh introduces other problems, such as the MPM's well known cell crossing instability. Another, less discussed, issue is the small cut instability, which is linked to the arbitrary interaction between the physical body and the background grid. There is the potential for degrees of freedom near the boundary of the physical body to have very small contributions from material points, resulting in a highly ill-conditioned linear system of equations. Similar issues are faced by the immersed/unfitted finite element community and this contribution explores the use of mesh aggregation within the MPM to stabilise the problem. The proposed Aggregated MPM (AgMPM) approach will be contrasted against ghost stabilisation as an alternative method to mitigate the small cut instability.