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This article concerns the asymptotic behavior of the following non-autonomous degenerate parabolic equation with singular perturbations defined on a bounded domain in $\mathbb{R}^n$, $$ \frac{\partial u}{\partial t}+\lambda u-\operatorname{div}(|\nabla u|^{p-2}\nabla u) -\varepsilon \operatorname{div}\Big(\big|\nabla \frac{\partial u}{\partial t}\big|^{p-2} \nabla \frac{\partial u}{\partial t}\Big) +f(x,t,u)=g(x,t), $$ where $\lambda$ is a positive constant, $p>2$ and $\varepsilon\in(0,1]$. The well-posedness and upper semicontinuity of pullback attractors are established for the problem without the uniqueness of solutions under singular perturbations.