Federated optimization under data heterogeneity presents a significant challenge, often leading to suboptimal model performance. While numerous methods aim to replicate the ideal performance of centralized training, they frequently fall short in highly heterogeneous settings. In this paper, we introduce HaFedHo, an adaptive objective rectification method that harmonizes local training with the ideal data-centralized objective, requiring minimal modifications to the standard federated learning framework. HaFedHo operates by first decoupling the centralized objective and then employing a dynamic Taylor series expansion to accurately estimate the global objective for each client. Our theoretical analysis shows that the estimation error provably converges to zero as training progresses. Furthermore, extensive experiments on real-world datasets demonstrate that HaFedHo surpasses state-of-the-art methods, including SCAFFOLD, MimeLite, and FedDyn, in both test accuracy and communication efficiency. Notably, HaFedHo maintains its superior performance even with a client participation rate as low as $0.2\%$ in severely heterogeneous environments.