A vortex lattice in a high-temperature superconductor (HTS) with an array of submicron magnetic dots on the surface has been studied by the Monte Carlo method within the framework of a three-dimensional model of a layered high-temperature superconductor. The adjustment of the vortex lattice to an array of magnetic points was observed — ordered states arising in the process of remagnetization — configurations of one, two, three, or more vortex filaments fixed to a single magnetic point. The occurrence of these configurations was accompanied by peaks on the magnetization curve. The influence of the HTS anisotropy on the adjustment of the vortex lattice is analyzed. The ordered configurations of the vortex lattice are also associated with the non-monotonic nature of the dependences of the superconductor critical current on the magnetic field. The influence of temperature, magnetic moment of points, and film thickness on the critical current is investigated. With an increase in temperature and a decrease in the magnetization of magnetic points, the maximum of the critical current shifts towards a lower field. The structure of vortex filaments in the inhomogeneous field of a magnetic point is analyzed in detail. The mechanism of the influence of ordered vortex configurations on magnetization and critical current is discussed.