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.
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