  Surface superconductivity has recently been observed on the (001) surface of
the topological crystalline insulator Pb$_{1-x}$Sn$_{x}$Te using point-contact
spectroscopy, and theoretically proposed to be of the chiral $p-$wave type. In
this paper, we closely examine the conditions for realizing a robust chiral
$p-$wave order in this system, rather than conventional $s$-wave
superconductivity. Further, within the $p$-wave superconducting phase, we
identify parameter regimes where impurity bound (Shiba) states depend crucially
on the existence of the chiral $p-$wave order, and distinguish them from other
regimes where the chiral $p-$wave order does exist but the impurity-induced
subgap bound states cannot be used as evidence for it. Such a distinction can
provide an easily realizable experimental test for chiral $p-$wave order in
this system. Notably, we have obtained exact analytical expressions for the
bound state wavefunctions in point defects, in the chiral $p-$wave
superconducting state, and find that instead of the usual $exponential$ decay
profile that characterizes bound states, these states decay as a $power-law$ at
large distances from the defect. As a possible application of our findings, we
also show that the zero-energy Shiba states in point defects possess an
internal SU(2) rotational symmetry which enables them to be useful as quantum
qubits.
