  We theoretically investigate the behavior of a mobile impurity immersed in a
one-dimensional quasi-periodic Fermi system with topological $p$-wave
superfluidity. This polaron problem is solved by using a standard variational
approach, the so-called Chevy ansatz. The polaron states are found to be
strongly affected by the strength of the quasi-disorder and the amplitude of
the $p$-wave pairing. We analyze the phase diagram of the polaron ground state
and find four phases: two extended phases, a weakly-localized phase and a
strongly-localized phase. It is remarkable that these polaron phases are
directly corresponding to the four distinct phases experienced by the
underlying background Fermi system. In particular, the weakly-localized polaron
phase corresponds to an intriguing critical phase of the Fermi system.
Therefore, the different phases of the background system can be unambiguously
probed by measuring the polaron properties via radio-frequency spectroscopy. We
also investigate the high-lying excited polaron states at an infinite
temperature and address the possibility of studying many-body localization
(MBL) of these states. We find that the introduction of $p$-wave pairing may
delocalize the many-body localized states and make the system easier to
thermalize. Our results could be observed in current state-of-the-art cold-atom
experiments.
