  We present the results of a high-cadence and high-precision radial velocity
(RV) monitoring of 3 late-type dwarf stars hosting long-period giants with
well-measured orbits, in order to search for short-period sub-Neptunes (SN, $M
\sin i < 30$ M$_\oplus$). Building on the results and expertise of our previous
studies, we carry out combined fits of our HARPS-N data with literature RVs,
using MCMC analyses and Gaussian Process regression. We then use the results of
our survey to estimate the frequency of sub-Neptunes in systems hosting
cold-Jupiters, $f(SN|CJ)$, and compare it with the frequency around field
M-dwarfs, $f(SN)$. We identify a new short-period low-mass planet orbiting GJ
328, GJ 328\,c, with $P_c = 241.8^{+1.3}_{-1.7}$ d and $M_c \sin i = 21.4^{+
3.4}_{- 3.2}$ M$_\oplus$. We moreover identify and model the chromospheric
activity signals and rotation periods of GJ 649 and GJ 849, around which no
additional planet is found. Then, taking into account also planetary system
around the previosuly-analyzed low-mass star BD-11 4672, we derive an estimate
of the frequencies of inner planets in such systems. In particular $f(SN|CJ) =
0.25^{+0.58}_{-0.07}$ for mini-Neptunes ($10$ M$_\oplus < M \sin i < 30$
M$_\oplus$, $P < 150$ d), marginally larger than $f(SN)$. For lower-mass
planets ($M \sin i < 10$ M$_\oplus$) instead $f(SN|CJ) <0.69$, compatible with
$f(SN)$. In light of the newly detected mini-Neptune, we find tentative
evidence of a positive correlation between the presence of those planets and
that of inner low-mass planets, $f(SN|CJ) > f(SN)$. This might indicate that
cold Jupiters have an opposite influence in the formation of inner sub-Neptunes
around late-type dwarfs as opposed to their solar-type counterparts, boosting
the formation of mini-Neptunes instead of impeding it.
