Broadband power-line communications (PLC) is considered as an enabling technology for the Internet-of-Things (IoT) systems. However, PLC faces some transmission issues as a power line was not designed to transmit communication data. Impulsive noise is one of the major issues in broadband PLC systems. In practice, impulsive noise may occur in bursts. The performance of the PLC systems degrades significantly under bursty impulsive noise environment. We propose to use a compressive sensing algorithm to detect and mitigate the bursty impulsive noise. In this paper, block sparse Bayesian learning (BSBL) algorithms are used to detect and estimate the bursty impulsive noise. The estimated impulsive noise is then subtracted from the contaminated signal. BSBL has relatively better performance than other block compressive sensing algorithms. We examine the performance of two types of BSBL algorithms, i.e. BSBL-bound optimization (BSBL-BO) and BSBL-expectation-maximization (BSBL-EM) under various bursty impulsive noise conditions. The results show that both algorithms have comparable performance in terms of bit error rate (BER). However, BSBL-EM gives better minimum square error (MSE) but much slower in the CPU processing time than BSBL-BO.