We have investigated the thermal-transport properties of the kagome antiferromagnet Cd-kapellasite (Cd-K). We find that a field suppression effect on the longitudinal thermal conductivity k_xx sets in below ~25 K, suggesting a large spin contribution k_xx^sp in k_xx. We also find clear thermal Hall signals in the spin liquid phase in all Cd-K samples. The magnitude of the thermal Hall conductivity k_xy shows a significant dependence on the samples scattering time. On the other hand, the temperature dependence of k_xy is similar in all Cd-K samples; k_xy shows a peak at almost the same temperature of the peak of the phonon thermal conductivity k_xy^ph which is estimated by k_xx at 15 T. These results indicate the presence of a dominant phonon thermal Hall k_xy^ph at 15 T. In addition to k_xy^ph, we find that the field dependence of k_xy at low fields turns out to be non-linear at low temperatures, concomitantly with the appearance of the field suppression of k_xx, indicating the presence of a spin thermal Hall k_xy^sp at low fields. Remarkably, by assembling the k_xx dependene of k_xy^sp data of other kagome antiferromagnets, we find that, whereas k_xy^sp stays a constant in the low-k_xx region, k_xy^sp starts to increase as k_xx does in the high-k_xx region. This k_xx dependence of k_xy^sp indicates the presence of both intrinsic and extrinsic mechanisms in the spin thermal Hall effect in kagome antiferromagnets. Furthermore, both k_xy^ph and k_xy^sp disappear in the antiferromagnetic ordered phase at low fields, showing that phonons alone do not exhibit the thermal Hall effect. A high field above ~7 T induces k_xy^ph, concomitantly with a field-induced increase of k_xx and the specific heat, suggesting a coupling of the phonons to the field-induced spin excitations as the origin of k_xy^ph.
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