  The latest Lyman-$\alpha$ forest data severely constrain the conventional
warm dark matter solution to small-scale issues in the cold dark matter
paradigm. It has been also reported that unconstrained astrophysical processes
may address the issues. In response to this situation, we revisit the decaying
dark matter solution to the issues, discussing possible signatures to
discriminate decaying dark matter from astrophysical processes as a solution to
small-scale issues. We consider an axinolike particle (ALPino) decaying into an
axionlike particle (ALP) and gravitino with the lifetime around the age of the
Universe. The ALPino mass is sub-PeV and slightly ($\Delta m/m\sim 10^{-4}$)
larger than the gravitino mass, and thus the dark matter abundance does not
alter virtually after the ALPino decays. On the other hand, the gravitino
produced from the ALPino decay obtains a kick velocity of $\sim 30 \,{\rm km /
s}$, which is sufficiently larger than a circular velocity of dwarf galaxies to
impact their dark matter distributions. The Lyman-$\alpha$ forest constraints
are relieved since only a small fraction ($\sim10$%) of dark matter experiences
the decay at that time. Decaying dark matter is thus promoted to a viable
solution to small-scale issues. The ALPino relic abundance is determined
predominantly by the decay of the lightest ordinary supersymmetric particle.
The monochromatic ALP emission from the ALPino decay is converted to $\sim 50
\,{\rm GeV}$ photon under the Galactic magnetic field. The morphology of the
gamma-ray flux shows a distinctive feature of the model when compared to
decaying dark matter that directly decays into photons. Once detected, such
distinctive signals discriminate the decaying dark matter solution to
small-scale issues from unconstrained astrophysical processes.
