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We aim to study a one-dimensional $p$-wave superconductor with quasiperiodicon-site potentials. A modified real-space-Pfaffian method is applied tocalculate the topological invariants. We confirm that the Majorana zero mode isprotected by the nontrivial topology the topological phase transition isaccompanied by the energy gap closing and reopening. In addition, wenumerically find that there are mobility edges which originate from thecompetition between the extended $p$-wave pairing and the localizedquasi-disorder. We qualitatively analyze the influence of superconductingpairing parameters and on-site potential strength on the mobility edge. Ingeneral, our work enriches the research on the $p$-wave superconducting modelswith quasiperiodic potentials.

Fate of Majorana zero modes by a modified real-space-Pfaffian method and mobility edges in a one-dimensional quasiperiodic lattice