Phase aberrations are introduced when focusing by a high-numerical aperture (NA) objective lens into refractive-index-mismatched (RIM) media. The axial focus position in these media can be adjusted through either optical remote-focusing or mechanical stage translation. Despite the wide interest in remote-focusing, no generalised control algorithm using Zernike polynomials has been presented that performs independent remote-focusing and RIM correction in combination with mechanical stage translation. In this work, we thoroughly review derivations that model high-NA defocus and RIM aberration. We show through both numerical simulation and experimental results that optical remote-focusing using an adaptive device and mechanical stage translation are not optically equivalent processes, such that one cannot fully compensate for the other without additional aberration compensation. We further establish new orthogonal modes formulated using conventional Zernike modes and discuss its device programming to control high-NA remote-focusing and RIM correction as independent primary modes in combination with mechanical stage translation for aberration-free refocusing. Numerical simulations are performed, and control algorithms are validated experimentally by fabricating graphitic features in diamond using direct laser writing.