Noise from rotating machinery in an enclosed space exhibits both narrowband and broadband characteristics. Hybrid active noise control (HANC) algorithms are usually used in this case to cancel both kinds of noise. However, sometimes also bursty noise occurs. In a mechanical system, for example, it may occur in every cycle if the gear wheel faces are damaged. Bursty noise has high amplitudes and short duration, and this may deteriorate the narrowband attenuation performance of the HANC method. To address this problem, a feedforward HANC algorithm with a sinusoidal residual error structure (HANC-SRE) is proposed in this paper. By applying the sinusoidal residual error (SRE) subsystem, the proposed HANC-SRE algorithm can decrease the error interference between different narrowband adaptive filters for different frequencies. The structure of the traditional HANC is then redesigned to use the broadband filter as the primary noise absorber and the narrowband filters as secondary ones to achieve better performance for bursty noise. Moreover, we also introduce a variable step-size updating method to improve the tracking performance of the system. Theoretical analysis of convergence conditions and instantaneous residual output error is performed to evaluate the performance of the HANC-SRE algorithm. Experimental results demonstrate that HANC-SRE achieves better tracking performance while also being more robust than conventional HANC methods.