  Conventional $n$-dimensional topological superconductors (TSCs) have
protected gapless $(n - 1)$-dimensional boundary states. In contrast to this,
second-order TSCs are characterized by topologically protected gapless $(n -
2)$-dimensional states with usual gapped $(n - 1)$-boundaries. Here, we study a
second-order TSC with a two-dimensional (2D) magnetic topological insulator
(TI) proximity-coupled to a high-temperature superconductor, where Majorana
bound states (MBSs) are localized at the corners of a square sample with gapped
edge modes. Due to the mirror symmetry of the hybrid system considered here,
there are two MBSs at each corner for both cases: d-wave and $s_{\pm}$-wave
superconducting pairing. We present the corresponding topological phase
diagrams related to the role of the magnetic exchange interaction and the
pairing amplitude. A detailed analysis, based on edge theory, reveals the
origin of the existence of MBSs at the corners of the 2D sample, which results
from the sign change of the Dirac mass emerging at the intersection of any two
adjacent edges due to pairing symmetry. Possible experimental realizations are
discussed. Our proposal offers a promising platform for realizing MBSs and
performing possible non-Abelian braiding in 2D systems.
