  A Yu-Shiba-Rusinov (YSR) state is a localized in-gap state induced by a
magnetic impurity in a superconductor. Recent experiments used an STM tip to
manipulate the exchange coupling between an Fe adatom and the ${\rm
FeTe}_{0.55}{\rm Se}_{0.45}$ superconductor possessing a $Z_2$ nontrivial band
structure with topological surface states. As the tip moves close to the single
Fe adatom, the energy of the in-gap state is modulated and exhibits two unusual
zero-energy crossings, which cannot be understood by coupling the magnetic
impurity to the superconducting topological surface Dirac cone. Here, we
numerically and analytically study the YSR states in superconductors with
nontrivial $Z_2$ bands and explain the origin of the two zero-energy crossings
as a function of the exchange coupling between the magnetic impurity and the
bulk states. We analyze the role of the topological surface states and compare
in-gap states to systems with trivial $Z_2$ bands. The spin-polarization of the
YSR states is further studied for future experimental measurement.
