  We provide a simple analysis of the big-bang nucleosynthesis (BBN)
sensitivity to the light dark matter (DM) generated by the thermal freeze-in
mechanism. It is shown that the ratio of the effective neutrino number shift
$\Delta N_{\nu}$ over the DM relic density $\omega\equiv \Omega h^2$, denoted
by $R_\chi\equiv\Delta N_\nu/\omega$, cancels the decaying particle mass and
the feeble coupling, rendering therefore a simple visualization of $\Delta
N_{\nu}$ at the BBN epoch in terms of the DM mass. This property drives one to
conclude that the shift with a sensitivity of $\Delta N_{\nu}\simeq
\mathcal{O}(0.1)$ cannot originate from a single warm DM under the
Lyman-$\alpha$ forest constraints. For the cold-plus-warm DM scenarios where
the Lyman-$\alpha$ constraints are diluted, the ratio $R_\chi$ can be
potentially used to test the thermal freeze-in mechanism in generating a small
warm component of DM and a possible excess at the level of $\Delta
N_{\nu}\simeq \mathcal{O}(0.01)$.
