We present theoretical studies of the effects of nonlinearity on the phonon transmission and energy transport through two-dimensional arrays of two-path anharmonic atomic defects in lattices and through two-path atomic-scale junctions. The two-path transmission results in destructive phonon interference and the appearance of antiresonances (transmission nodes) in transmission spectrum. We demonstrate the appearance of the transmission antiresonance with the few-particle nanostructures that allow to model both linear and nonlinear phonon antiresonances. The anharmonicity of the considered atomic-scale systems is in the interatomic bonds. The proposed nonlinear approach describes the generation of the higher wave harmonics in result of the interaction of lattice wave with nonlinear two-path atomic defect or atomic-scale junction. We derive the full system of nonlinear algebraic equations, which describe the transmission through nonlinear two-path atomic defects with an account for the generation of second and third harmonics. We obtain the expressions for the coefficients of energy transmission through and reflection from the nonlinear atomic-scale systems. We show that the quartic interatomic nonlinearity in the atomic defect shifts the antiresonance frequency in the direction, determined by the sign of the nonlinear coefficient. The quartic nonlinearity also enhances in general the transmission of high-frequency phonons due to third harmonic generation and propagation. The effects of the quartic nonlinearity on phonon transmission and propagation are described for the two-path atomic defects with different topology. We also model the transmission through the nonlinear two-path atomic defects by launching the phonon wave packet, for which the proper amplitude normalization is proposed and implemented. We show that the cubic interatomic nonlinearity in the atomic defect red shifts in general the antiresonance frequency independently of the sign of the nonlinear coefficient, and the equilibrium interatomic distances (bond lengths) in the atomic defect are changed by the incident phonon due to cubic interatomic nonlinearity. In the systems with the cubic nonlinearity, a new narrow transmission resonance on the background of a broad antiresonance can emerge, which we relate with the opening of the additional transmission channel for phonon second harmonic through the nonlinear defect atoms. The considered effects of interference and nonlinearity are applied to the modeling of phonon energy transport in cubic lattices of Si atoms with two-dimensional sparse arrays of Ge atoms and of phonon energy transport through two-path single-molecule junctions. The presented results contribute to the better understanding and detailed modeling of the interplay between the interference and nonlinearity in phonon propagation through and scattering in two-dimensional arrays of two-path anharmonic atomic defects and in atomic-scale junctions with different topology.