Probing CP-violating Higgs-Gauge couplings with Higgsstrahlung at $e^-e^+$ collider

We investigate the sensitivity of a future high-luminosity $e^-e^+$ collider operating at $\sqrt{s}=250~\text{GeV}$ to CP-violating and CP-conserving anomalous $hVV$ interactions via the Higgsstrahlung process. The effects of new physics are parameterized in the Standard Model Effective Field Theory~(SMEFT) framework through six dimension-6 operators modifying the $hVV$ vertices. Using polarized beams and exploiting polarization and spin correlation asymmetries reconstructed from Higgs decay products, we perform a comprehensive analysis across the three dominant decay modes, $h\to b\bar{b}$, $WW^\star$, and $ZZ^\star$. The $h\to WW^\star$ channel exhibits the highest sensitivity to $\mathcal{O}_{HW}$ and $\mathcal{O}_{H\widetilde{W}}$, while the $b\bar{b}$ mode constrains the remaining operators with high statistical precision. Sensitivity studies incorporating luminosity scaling and systematic uncertainties show that the projected bounds improve significantly with increasing integrated luminosity, but saturate once experimental systematics exceed the few-percent level. These results highlight the crucial role of spin-based observables and beam polarization in achieving sub-percent precision on SMEFT coefficients at future lepton colliders.