Dipole mixed modes observed in the oscillation pattern of red giant stars probe the radiative regions in the stellar core. Oscillation spectra of helium-core-burning stars sometimes show extra peaks that remain unexplained by the dipole mixed-mode pattern expected from the coupling of a radiative cavity in the stellar core and a pressure cavity in the stellar envelope. We use the asymptotic expansion developed for a multi-cavity star in order to characterize these extra peaks. The analytical resonance condition of the multi-cavity gravito-acoustic modes, with two inner gravity cavities and an outer pressure cavity, helps us explain that the apparent extra peaks are dipole mixed modes that follow the three-cavity oscillation pattern. The derivation of the two asymptotic period spacings associated with the two distinct regions in the radiative core provides an estimate of the full radiative cavity. Our results provide new constraints for analyzing the overshoot or mixing in the core of helium-core-burning stars. An important structure discontinuity inside the radiative core may explain the larger than expected observed period spacings.