This study investigates the vulnerability of target signals to co-channel Linear Frequency Modulation (LFM) interference in automotive Frequency-Modulated Continuous-Wave (FMCW) radar systems. It analyzes the limitations of conventional adaptive noise cancellation (ANC) techniques, particularly slow convergence and performance degradation under intense interference. To address these issues, an improved ANC algorithm is proposed. The method generates reference signals through single-channel self-delay processing and adopts a joint optimization framework for weight adaptation, which integrates normalized variable-step-size Least Mean Squares (LMS) adaptation with a leakage factor. Notably, the algorithm achieves robust performance in high-interference scenarios without requiring additional hardware or complex signal transformations. Simulation results verify that the proposed algorithm significantly improves the signal-to-interference-plus-noise ratio (SINR) preserves signal fidelity, and enhances detection probability under strong LFM interference.