  This work studies a two-time-scale functional system given by two
jump-diffusions under the scale separation by a small parameter $\varepsilon
\rightarrow 0$. The coefficients of the equations that govern the dynamics of
the system depend on the segment process of the slow variable (responsible for
capturing delay effects on the slow component) and on the state of the fast
variable. We derive a moderate deviations principle for the slow component of
the system in the small noise limit using the weak convergence approach. The
rate function is written in terms of the averaged dynamics associated to the
multi-scale system. The core of the proof of the moderate deviations principle
is the establishment of an averaging principle for the controlled processes
associated to the slow variable in the framework of the weak convergence
approach. The controlled version of the averaging principle for jump
multi-scale diffusions relies on the classical Khasminkii's technique.
