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Multimode interference and multipolar interplay govern functionalities of optical nanoresonators and nonlinear nanoantennas. Recently, excitation of the high-quality supercavity modes (quasi-BIC states) in individual subwavelength dielectric particles has been predicted to boost the nonlinear frequency conversion at the nanoscale. Here, we put forward the multipolar model which captures the physics behind linear and nonlinear response driven by such high-$Q$ modes in nanoresonators. We show that formation of the quasi-BIC state in the AlGaAs nanodisk can be understood through multipolar transformations of coupled leaky modes. In particular, the hybridized axially symmetric TE-polarized modes can be viewed as superpositions of multipoles, with a basis of parent multipoles constituted mainly by magnetic dipoles and octupole. The quasi-BIC point in the parameter space appears where dipolar losses are totally suppressed. The efficient optical coupling to this state is implemented via azimuthally polarized beam illumination matching its multipolar origin. We establish a one-to-one correspondence between the standard non Hermitian coupled-mode theory and multipolar models that enables transparent interpretation of scattering characteristics. Using our approach, we derive the multipolar composition of the generated second-harmonic radiation from the AlGaAs nanodisk and validate it with full-wave numerical simulations. Back-action of the second-harmonic radiation onto the fundamental frequency is taken into account in the coupled nonlinear model with pump depletion. A hybrid metal-dielectric nanoantenna is proposed to augment the conversion efficiency up to tens of per cent and actualize the nonlinear parametric downconversion. Our findings delineate the in-depth conceptual framework and novel promising strategies in the design of functional elements for nonlinear nanophotonics applications.