Large language models exhibit strong reasoning capabilities, yet often rely on shortcuts such as surface pattern matching and answer memorization rather than genuine logical inference. We propose Shortcut-Aware Reasoning Training (SART), a gradient-aware framework that detects and mitigates shortcut-promoting samples via ShortcutScore and gradient surgery. Our method identifies shortcut signals through gradient misalignment with validation objectives and answer-token concentration, and modifies training dynamics accordingly. Experiments on controlled reasoning benchmarks show that SART achieves +16.5% accuracy and +40.2% robustness over the strongest baseline, significantly improving generalization under distribution shifts. Code is available at: https://github.com/fuyanjie/short-cut-aware-data-centric-reasoning.