Summary: In this paper, we prove an \(L^2\) extension theorem for holomorphic sections of holomorphic line bundles equipped with singular metrics on weakly pseudoconvex Kähler manifolds. Furthermore, in our \(L^2\) estimate, optimal constants corresponding to variable denominators are obtained. As applications, we prove an \(L^q\) extension theorem with an optimal estimate on weakly pseudoconvex Kähler manifolds and the log-plurisubharmonicity of the fiberwise Bergman kernel in the Kähler case.