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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 withwell-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 previousstudies, 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 ofour survey to estimate the frequency of sub-Neptunes in systems hostingcold-Jupiters, $f(SN|CJ)$, and compare it with the frequency around fieldM-dwarfs, $f(SN)$. We identify a new short-period low-mass planet orbiting GJ328, 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 chromosphericactivity signals and rotation periods of GJ 649 and GJ 849, around which noadditional planet is found. Then, taking into account also planetary systemaround the previosuly-analyzed low-mass star BD-11 4672, we derive an estimateof 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-massplanets ($M \sin i < 10$ M$_\oplus$) instead $f(SN|CJ) <0.69$, compatible with$f(SN)$. In light of the newly detected mini-Neptune, we find tentativeevidence of a positive correlation between the presence of those planets andthat of inner low-mass planets, $f(SN|CJ) > f(SN)$. This might indicate thatcold Jupiters have an opposite influence in the formation of inner sub-Neptunesaround late-type dwarfs as opposed to their solar-type counterparts, boostingthe formation of mini-Neptunes instead of impeding it.

The GAPS programme at TNG. XLVI. Deep search for low-mass planets in late-dwarf systems hosting cold Jupiters