This paper develops a physics-guided graph network to enhance the robustness of distribution system state estimation (DSSE) against anomalous real-time measurements, as well as a deep auto-encoder (DAE)-based detector and a Gaussian process-aided residual learning (GARL) to deal with challenges arising from topology changes. A global-scanning jumping knowledge network (GSJKN) is first designed to establish the regression rule between the measurement data and state variables. The structural information of distribution system (DS) and a global-scanning module are incorporated to guide the propagation of scarce measurements in the graph topology, contributing to valid estimation precision in sparsely measured DSs. To monitor the topology changes of the network, a DAE network is employed to learn an efficient representation of the measurements of the system under a certain topology, which can achieve online monitoring of the network structure by observing the variation tendency of the reconstruction error. When the topology change occurs, a Gaussian process with a composite kernel is applied to the modeling of the pre-trained GSJKN residual to adapt to the new topology. The embedding of the physical structural knowledge enables the proposed GSJKN method to restore the missing/noisy values utilizing the adjacent measurements, which enhances the robustness to typical data acquisition errors. The adopted DAE network and special GARL-based transfer method further allow the DSSE method to rapidly detect and adapt to the topology change, as well as achieve effective quantification of the estimation uncertainties. Comparative tests on balanced and unbalanced systems demonstrate the accuracy, robustness, and adaptability of the proposed DSSE method.