Traffic flow prediction plays a crucial role in alleviating traffic congestion and enhancing transport efficiency. While combining graph convolution networks with recurrent neural networks for spatial-temporal modeling is a common strategy in this realm, the restricted structure of recurrent neural networks limits their ability to capture global information. For spatial modeling, many prior studies learn a graph structure that is assumed to be fixed and uniform at all time steps, which may not be true. This paper introduces a novel traffic prediction framework, Global-Aware Enhanced Spatial-Temporal Graph Recurrent Network (GA-STGRN), comprising two core components: a spatial-temporal graph recurrent neural network and a global awareness layer. Within this framework, three innovative prediction models are formulated. A sequence-aware graph neural network is proposed and integrated into the Gated Recurrent Unit (GRU) to learn non-fixed graphs at different time steps and capture local temporal relationships. To enhance the model's global perception, three distinct global spatial-temporal transformer-like architectures (GST^2) are devised for the global awareness layer. We conduct extensive experiments on four real traffic datasets and the results demonstrate the superiority of our framework and the three concrete models.