  We study the time evolution of the Fr\"ohlich Hamiltonian in a mean-field
limit in which many particles weakly couple to the quantized phonon field.
Assuming that the particles are initially in a Bose-Einstein condensate and
that the excitations of the phonon field are initially in a coherent state we
provide an effective dynamics which approximates the time evolved many-body
state in norm, provided that the number of particles is large. The
approximation is given by a product state which evolves according to the
Landau-Pekar equations and which is corrected by a Bogoliubov dynamics. In
addition, we extend the results from [Arch. Ration. Mech. Anal. 240, 383-417
(2021)] about the approximation of the time evolved many-body state in
trace-norm topology to a larger class of many-body initial states with an
improved rate of convergence.
