Probing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si38.svg"> <mml:mrow> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>η</mml:mi> <mml:mrow> <mml:mo>(

Motivated by the successful application of Heavy Quark Effective Field Theory in describing decays from heavy to light mesons, this work explores its applicability to the semileptonic decays of charmed mesons. So in this paper we investigate the $D_s^+\to η^{(\prime)} \ell^+ ν_\ell$ transition form factors using the light-cone sum rules approach within the framework of heavy-quark effective field theory. To address the large uncertainties arsing from the $η^{(\prime)}$-meson twist-3 distribution amplitudes, we employ the right-handed chiral correlation function. By applying the converging simplified series expansion method, we extrapolate the form factors to the entire physical $q^2$-region. Our analysis yields the branching fractions precise predictions for semi-leptonic decays $D_s^+\to η^{(\prime)}\ell^+ν_\ell$ with : $\mathcal{B}(D_s^+\toη\ell^+ν_\ell)=2.300^{+0.230}_{-0.227}\%$ ($\ell = e$) and $2.249_{-0.206}^{+0.209}\%$ ($\ell = μ$); $\mathcal{B}(D_s^+\toη^\prime \ell^+ν_\ell)=0.861^{+0.095}_{-0.093}\%$ ($\ell = e$) and $0.821^{+0.082}_{-0.080}\%$ ($\ell = μ$). The derived lepton flavor universality ratios $R^η_{μ,e}=0.977^{+0.008}_{-0.006}$ and $R^{η^{\prime}}_{μ,e} = 0.953^{+0.011}_{-0.009}$ are consistent with lasted BESIII experimental measurements. Additionally, the forward-backward asymmetry parameters $\langle \mathcal{A}^η_{\rm FB}\rangle=-0.034^{+0.003}_{-0.003}$ and $\langle \mathcal{A}^{η^\prime}_{\rm FB}\rangle=-0.073^{+0.007}_{-0.008}$ suggest that no significant violation of lepton flavor universality in this decay process.