  We investigate new physics with light-neutral mediators through coherent
elastic neutrino-nucleus scattering (CE$\nu$NS) at low energies. These
mediators, with a mass of less than $1$ GeV, are common properties for
extensions of the Standard Model (SM). We consider general scalar, vector, and
tensor interactions allowed by Lorentz invariance and involve universal light
mediators accordingly. In addition, we study an additional vector gauge boson
with an associated $U(1)'$ gauge group for a variety of models including
$U(1)_{B-L}$, $U(1)_{B-3L_e}$, $U(1)_{B-3L_\mu}$, and $U(1)_{B-3L_\tau}$. These
models differ in the fermion charges, which determine their contributions
within the CE$\nu$NS process. The effects of each model are investigated by
embedding them in the SM process using solar neutrino flux. We derive new
limits on the coupling-mass plane of these models from the latest CDEX-10 data.
We also present projected sensitivities involving the future experimental
developments for each model. Our results provide more stringent constraints in
some regions, compared to previous works. Furthermore, the projected
sensitivities yield an improvement of up to one order of magnitude.
