Neutrino-nuclear responses in the form of squares of nuclear matrix elements, NMEs, are crucial for studies of neutrino-induced processes in nuclei. In this work we investigate magnetic hexadecapole (M4) NMEs in medium-heavy nuclei. The experimentally derived NMEs, $M_{EXP}$(M4), deduced from observed M4 $\gamma$ transition half-lives are compared with the single-quasiparticle (QP) NMEs, $M_{QP}$(M4), and the microscopic quasiparticle-phonon model (MQPM) NMEs $M_{MQPM}$(M4). The experimentally derived M4 NMEs are found to be reduced by a coefficient $k\approx 0.29$ with respect to $M_{QP}$(M4) and by $k\approx 0.33$ with respect to $M_{MQPM}$(M4). The M4 NMEs are reduced a little by the quasiparticle-phonon correlations of the MQPM wave functions but mainly by other nucleonic and nonnucleonic correlations which are not explicitly included in the MQPM. The found reduction rates are of the same order of magnitude as those for magnetic quadrupole $\gamma$ transitions and Gamow-Teller (GT) and spin-dipole (SD) $\beta$ transitions. The impacts of the found reduction coefficients on the magnitudes of the NMEs involved in astroneutrino interactions and neutrinoless double beta decays are discussed.