  Cosmologies with Light Massive Relics (LiMRs) as a subdominant component of
the dark sector are well-motivated from a particle physics perspective, and can
also have implications for the $\sigma_8$ tension between early and late time
probes of matter clustering. The effects of LiMRs on the Cosmic Microwave
Background (CMB) and structure formation on large (linear) scales have been
investigated extensively. In this paper, we initiate a systematic study of the
effects of LiMRs on smaller, nonlinear scales using cosmological $N$-body
simulations; focusing on quantities relevant for photometric galaxy surveys.
For most of our study, we use a particular model of nonthermal LiMRs but the
methods developed easily generalize to a large class of models of LiMRs -- we
explicitly demonstrate this by considering the Dodelson-Widrow form of the
velocity distribution. We find that, in general, the effects of LiMR on small
scales are distinct from those of a $\Lambda$CDM universe, even when the value
of $\sigma_8$ is matched between the models. We show that weak lensing
measurements around massive clusters, between $\sim 0.1 h^{-1}$Mpc and $\sim 10
h^{-1}$Mpc, should have sufficient signal-to-noise in future surveys to
distinguish between $\Lambda$CDM and LiMR models that are tuned to fit both CMB
data and large (linear) scale structure data at late times. Furthermore, we
find that different LiMR cosmologies which are indistinguishable by
conventional linear probes can be distinguished by these probes if their
velocity distributions are sufficiently different. LiMR models can, therefore,
be best tested and constrained by jointly analyzing data from CMB and late-time
structure formation on both large \textit{and} small scales.
