On the surface of an s-wave superconductor without spin-orbit coupling, the combination of magnetic flux and antiferromagnetic order induced by adsorbed magnetic atoms can give rise to higher-order topological superconductivity. However, in realistic magnetic lattices, the magnetic order often deviates from the ideal antiferromagnetic configuration due to various influencing factors, resulting in different types of distortions. To this end, this paper introduces ferromagnetic order and spiral magnetic order as distortion factors into the antiferromagnetic model, systematically investigates the higher-order topological superconducting properties of this model, and thoroughly analyzes the physical origin of the higher-order topological states. We employ the quadrupole moment topological invariant to demonstrate that the boundary corner states are higher-order topological corner states. Meanwhile, we understand the formation mechanism of the topological corner states from the perspective of the low-energy boundary effective Hamiltonian. The results show that antiferromagnetic order is a prerequisite for realizing higher-order topological superconductivity; although ferromagnetic order renormalizes the band structure, it does not destroy the higher-order topological gap. In contrast, the introduction of a spiral component causes a tilting distortion of the energy bands. Once the angle between adjacent magnetic moments increases to a critical value, the energy gap completely closes and the topological states vanish accordingly.