Federated learning, as a distributed privacy protection technology, is widely used to solve the data island problem. To address the straggler problem of synchronous training, semi-asynchronous federated learning is proposed. Latency and data heterogeneity seriously affect the efficiency of semi-asynchronous federated learning. To this end, we propose a semi-asynchronous FL framework FAGA2 based on guided client selection and adaptive update aggregation, which includes three modules. The first module is to guide the client selection, which selects the clients to participate in the training based on the outdated degree and update quality to cope with delay and heterogeneity. The second module is adaptive update aggregation, which optimizes local updates by minimizing the similarity with historical global updates, and evaluates weights to aggregate optimized local updates based on delay and update importance to mitigate bias. The third module is L2-norm amplification adjustment, which takes the average L2-norm of local updates as the L2-norm of global updates to adjust the aggregation bias. The effectiveness of the proposed method is verified on six synthetic and real datasets of different sizes. Experimental results show that compared with the baseline method, FAGA2 reduces the time to reach the target training accuracy by \(13.6\%\sim 72.5\%\) , and improves the time accuracy by \(1.1\times \sim 3.5\times \) while maintaining or improving the accuracy.