We investigate the influence of a solid core and of the cross-link density on the compression of microgel particles at oil water interfaces by means of compression isotherms and computer simulations. We investigate particles with different morphology, namely core shell particles containing a solid silica core surrounded by a cross-linked polymer shell of poly(N-isopropylacrylamide), and the corresponding hollow microgels where the core was dissolved. The polymer shell contains different amounts of cross-linker. The compression isotherms show that the removal of the core leads to an increase of the surface pressure at low compression, and the same effect can be observed when the polymer cross-link density is decreased. Low cross-link density and a missing core thus facilitate spreading of the polymer chains at the interface and, at high compression, hinder the transition to close hexagonal packing. Furthermore, the compression modulus only depends on the cross-link density at low compression, and no difference can be observed between the core shell particles and the corresponding hollow microgels. It is especially remarkable that a low cross-link density leads to a high compression modulus at low compression, while this behavior is reversed at high compression. Thus, the core does not influence the particle behavior until the polymer shell is highly compressed and the core is directly exposed to the pressure. This is related to an enhanced spreading of polymer chains at the interface and thus high adsorption energy. These conclusions are fully supported by computer simulations which show that the cross-link density of the polymer shell defines the degree of deformation at the interface. Additionally, the core restricts the spreading of polymer chains at the interface. These results illustrate the special behavior of soft microgels at liquid interfaces.