  We calculate the quasiparticle dispersion and spectral weight of the
quasiparticle that results when a hole is added to an antiferromagnetically
ordered CuO$_2$ plane of a cuprate superconductor. We also calculate the magnon
contribution to the quasiparticle spectral function. We start from a multiband
model for the cuprates considered previously [Nat. Phys. \textbf{10}, 951
(2014)]. We map this model and the operator for creation of an O hole to an
effective one-band generalized $t-J$ model, without free parameters. The
effective model is solved using the state of the art self-consistent Born
approximation. Our results reproduce all the main features of experiments. They
also reproduce qualitatively the dispersion of the multiband model, giving
better results for the intensity near wave vector $(\pi,\pi)$, in comparison
with the experiments. In contrast to what was claimed in [Nat. Phys.
\textbf{10}, 951 (2014)], we find that spin fluctuations play an essential role
in the dynamics of the quasiparticle, and hence in both its weight and
dispersion.
