  We consider a class of weakly asymmetric continuous microscopic growth models
with polynomial smoothing mechanisms, general nonlinearities and a Poisson type
noise. We show that they converge to the KPZ equation after proper rescaling
and re-centering, where the coupling constant depends nontrivially on all
details of the smoothing and growth mechanisms in the microscopic model. This
confirms some of the predictions in [HQ18].
  The proof builds on the general discretisation framework of regularity
structures ([EH19]), and employs the idea of using the spectral gap inequality
to control stochastic objects as developed and systematised in [LOTT21, HS24],
together with a new observation on structures of the Malliavin derivatives in
our situation.
