Usually, the structure of paraxial light beams is characterized by the intensity associated with distribution of the longitudinal energy flow density (Poynting momentum) across the transverse plane. In this work, special attention is paid to the distribution of internal energy flows described by the transverse Poynting momentum (TPM) components. This approach discloses additional polarization-dependent features of the vector beam transformations; in application to the edge diffraction of a circularly polarized (CP) Gaussian beam, it reveals the helicity-dependent asymmetry of the diffracted-field TPM profile characterized by the shifts of the TPM singularity, maximum, etc. These phenomena are confirmed experimentally and interpreted in terms of the spin-orbit interaction (SOI) and spin Hall effect of light. In contrast to the known SOI manifestations in the CP beams' diffraction originating from the small longitudinal component of a paraxial field, the new TPM-related effects stem from the transverse field components and are thus much higher in magnitude.