  Recent experimental investigations of arrays of magnetic atoms deposited on
top of a superconductor have opened a new chapter in the search of topological
superconductivity. We generalize the microscopic model derived by Pientka et
al. [Phys. Rev. B \textbf{88}, 155420 (2013)] to accommodate the effects of
finite supercurrent in the host material. Previously it was discovered that
helical chains with nonplanar textures are plagued by a gapless phase. We show
that by employing supercurrent it is possible to tune the chain from the
gapless phase to the topological gapped phase. It is also possible to tune the
chain between the trivial and the topological gapped phase, the size of which
may be dramatically increased due to supercurrent. For planar textures
supercurrent mainly contributes to proliferation of the gapless phase. Our
predictions, which can be probed in STM experiments, are encouraging for
observation and manipulation of Majorana states.
