Correction to $CP$-asymmetry in $\mathrm{{\rm Y}}(4S)$ decays due to the admixture of $B^0{\bar{B}}^0$ in a $C$-even state

$BB$ pairs from $\mathrm{{\rm Y}}(4S)\to BB$ decays are in $C$-odd state. However, there is a small admixture of the $C$-even state and it modifies the time dependent $CP$-asymmetry. The $C$-even component appears due to soft photon emission, breaking the pure $C$-odd nature of the initial state. Using the two-particle wave function formalism, we derive analytical expressions for $CP$-asymmetry in both pure and mixed $C$-parity states. We emphasize the strong energy dependence of the admixture magnitude.