In this paper, we study a structural acoustics model with supercritical source and nonlinear damping. The model consists of a wave equation defined on a bounded domain which is strongly coupled with a Berger plate equation acting on the elastic wall. The aim of the paper is to remove the strong condition $u\in L^{\mathrm{\infty }}({\mathbb{R}}^{+};L^{\frac{3}{2}}(\mathrm{\Omega }))$ for $m_1>5$ in the work ^[1] to obtain a weaker energy decay, and give a logarithmic decay rate under a weak assumption on the initial data.