In this paper (arXiv:2206.12717) we consider an N-dimensional multiparametric generalization of the classical Zernike system. We prove that it always provides a superintegrable system by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta. Such generic results are not only applied to the Euclidean plane, but also to the sphere and the hyperbolic plane. In the latter curved spaces, the Hamiltonian is expressed in geodesic polar coordinates showing that such a new superintegrable Hamiltonian can be regarded as a superposition of the isotropic 1:1 curved (Higgs) oscillator with even-order anharmonic curved oscillators plus another superposition of higher-order momentum-dependent potentials. Furthermore, the Racah algebra determined by the constants of the motion is also studied, giving rise to a (2N−1)th-order polynomial algebra. As a byproduct, the Hamiltonian is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that this new Hamiltonian (and so the Zernike system as well) is endowed with a Poisson 𝔰𝔩(2,ℝ)-coalgebra symmetry which would allow for further possible generalizations that are also discussed.