We study the Langevin dynamics corresponding to the $\nabla \varphi$-interface model with a degenerate convex interaction potential satisfying a polynomial growth assumption. Following the work of the author and Armstrong, we interpret these Langevin dynamics as a nonlinear parabolic equation forced by white noise and apply homogenization methods to derive a quantitative hydrodynamic limit. This result quantifies and extends to a class of degenerate convex potentials the seminal result of Funaki and Spohn. In order to handle the degeneracy of the potential, we make use of the notion of moderated environment originally introduced by Mourrat and Otto and further developed by Biskup and Rodriguez to study the properties of solutions of parabolic equations with degenerate coefficients (and of the corresponding random walks).