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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], and provides a first example of Hairer-Quastel type with both a generic nonlinearity (non-polynomial) and a non-Gaussian noise. 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 the specific structure of the (discrete) Malliavin derivatives in our situation. This structure enables us to reduce the control of mixed $L^p$ spacetime norms (of arbitrarily large $p$) by certain $L^2$-norms in spacetime., Comment: 62 pages. Significantly revised and expanded to improve exposition and explanation of calculations