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 $mathcal{B}_{D_s^+ rightarrow eta e^{+} u_e }$ = $(2.323pm0.063_{rm stat}pm0.063_{rm syst})%$ and $mathcal{B}_{D_s^+ rightarrow eta^{prime} e^{+} u_e}$ = $(0.824pm0.073_{rm stat}pm0.027_{rm syst})%$ via a tagged analysis technique, where one $D_s$ is fully reconstructed in a hadronic mode. Combining these measurements with previous BESIII measurements of $mathcal{B}_{D^+toeta^{(prime)} e^{+} u_e}$, the $eta-eta^prime$ mixing angle in the quark flavour basis is determined to be $phi_{rm P} = (40.1pm2.1_{rm stat}pm0.7_{rm syst})^circ$. From the first measurements of the dynamics of $D^+_sto eta^{(prime)}e^+ u_e$ 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 $f^{eta}_+(0)|V_{cs}| = 0.4455pm0.0053_{rm stat}pm0.0044_{rm syst}$ and $f^{eta^{prime}}_+(0)|V_{cs}| = 0.477pm0.049_{rm stat}pm0.011_{rm syst}$.
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