Despite their widespread use, purely data-driven methods often suffer from overfitting, lack of physical consistency, and high data dependency, particularly when physical constraints are not incorporated. This study introduces a novel data assimilation approach that integrates Graph Neural Networks (GNNs) with optimisation techniques to enhance the accuracy of mean flow reconstruction, using Reynolds-Averaged Navier-Stokes (RANS) equations as a baseline. The method leverages the adjoint approach, incorporating RANS-derived gradients as optimisation terms during GNN training, ensuring that the learned model adheres to physical laws and maintains consistency. Additionally, the GNN framework is well-suited for handling unstructured data, which is common in the complex geometries encountered in Computational Fluid Dynamics (CFD). The GNN is interfaced with the Finite Element Method (FEM) for numerical simulations, enabling accurate modelling in unstructured domains. We consider the reconstruction of mean flow past bluff bodies at low Reynolds numbers as a test case, addressing tasks such as sparse data recovery, denoising, and inpainting of missing flow data. The key strengths of the approach lie in its integration of physical constraints into the GNN training process, leading to accurate predictions with limited data, making it particularly valuable when data are scarce or corrupted. Results demonstrate significant improvements in the accuracy of mean flow reconstructions, even with limited training data, compared to analogous purely data-driven models.