In this paper, a class of spatially coupled generalized low-density parity-check (SC-GLDPC) codes with minimum coupling width is introduced to mitigate the Class-A impulsive noise. Codes with minimum coupling width are beneficial to simple code structure, short window decoding delay, and small rate-loss. The SC-GLDPC code is constructed by coupling the multiple identical generalized low-density parity-check (GLDPC) codes. Decoding thresholds over impulsive channels are derived by a recursive extrinsic information transfer function approach and are further estimated by the fast prediction approach. An edge spreading optimization method is proposed to design SC-GLDPC codes with minimum coupling width. The simulation results show that the optimized SC-GLDPC codes with minimum coupling width can show capacity-approaching property over impulsive channels. Benefitting from spatial coupling and generalized check nodes the SC-GLDPC codes show better bit error rate performance over impulsive channels, compared with GLDPC codes and spatially coupled low-density parity-check (SC-LDPC) codes.