  Using $e^+e^-$ annihilation data corresponding to an integrated luminosity of
3.19\,fb$^{-1}$ collected at a center-of-mass energy of 4.178~GeV with the
BESIII detector, we measure the absolute branching fractions
$\mathcal{B}_{D_s^+ \rightarrow \eta e^{+} \nu_e }$ = $(2.323\pm0.063_{\rm
stat}\pm0.063_{\rm syst})\%$ and $\mathcal{B}_{D_s^+ \rightarrow \eta^{\prime}
e^{+} \nu_e}$ = $(0.824\pm0.073_{\rm stat}\pm0.027_{\rm syst})\%$ via a tagged
analysis technique, where one $D_s$ is fully reconstructed in a hadronic mode.
Combining these measurements with previous BESIII measurements of
$\mathcal{B}_{D^+\to\eta^{(\prime)} e^{+} \nu_e}$, the $\eta-\eta^\prime$
mixing angle in the quark flavour basis is determined to be $\phi_{\rm P} =
(40.1\pm2.1_{\rm stat}\pm0.7_{\rm syst})^\circ$. From the first measurements of
the dynamics of $D^+_s\to \eta^{(\prime)}e^+\nu_e$ decays, the products of the
hadronic form factors $f_+^{\eta^{(\prime)}}(0)$ and the
Cabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ are determined with
different form factor parameterizations. For the two-parameter series
expansion, the results are $f^{\eta}_+(0)|V_{cs}| = 0.4455\pm0.0053_{\rm
stat}\pm0.0044_{\rm syst}$ and $f^{\eta^{\prime}}_+(0)|V_{cs}| =
0.477\pm0.049_{\rm stat}\pm0.011_{\rm syst}$.
