  We present a comprehensive $1/S$ study of the field-induced dynamical
properties of the nearest-neighbor $XXZ$ antiferromagnet on a honeycomb lattice
using the formalism of the nonlinear spin-wave theory developed for this model.
The external magnetic field controls spin frustration in the system and induces
non-collinearity of the spin structure, which is essential for the two-magnon
decay processes. Our results include an intriguing field-evolution of the
regions of the Brillouin zone wherein decays of spin excitations are prominent,
a detailed classification of the decay channels involving magnons from both
excitation branches, and a thorough analysis of the singularities in the magnon
spectra due to coupling to the two magnon continuum, all of which are
illustrated for several field and anisotropy values. We highlight a number of
features related to either the non-Bravais nature of the lattice, or the
existence of the Dirac-like points in the spectrum. In addition, the asymptotic
behavior of the decay rates near high-symmetry points is analyzed in detail.
The inelastic neutron-scattering spin-spin structure factor is obtained in the
leading $1/S$ order and is shown to exhibit qualitatively distinct fingerprints
of the decay-induced magnon dynamics such as quasiparticle peaks broadened by
decays and strong spectral weight redistribution.
