We study the singular stochastic wave equation on T 2 , with a cubic nonlinearity and Gaussian rough 'Matérn' forcing (a Fourier multiplier of order α > 0 applied to space-time white noise) and establish local well-posedness for α < 3 8. This extends [GKO18] beyond white noise and strengthens the quadratic-case result [OO21] (α < 1 2). Our argument develops new trilinear estimates in Bourgain spaces together with case-specific cubic counting estimates.