For the sparse system identification under non-Gaussian noise, the lp-norm-penalized proportionate affine projection generalized maximum correntropy (LP-PAPGMC) algorithm is proposed; This algorithm integrates the robustness of generalized maximum correntropy against impulsive noise and the adaptability of affine projection for handling highly correlated input signals, along with the enhancements in performance for sparse systems provided by the proportionate matrix and lp-norm-penalized constraint; Considering the inherent trade-off between convergence speed and steady-state bias due to the use of a constant kernel width for weight updates in the LP-PAPGMC algorithm, a variable kernel width strategy is utilized to dynamically adjust the kernel width of the algorithm, leading to the development of the LP-VKWPAPGMC algorithm; By identifying systems of varying sparsity under Gaussian noise and mixed Gaussian noise conditions, it has been validated that the LP-PAPGMC algorithm and its variable kernel width version exhibit faster convergence rates and lower steady-state biases compared to other related affine projection algorithms; In acoustic echo cancellation scenarios under mixed Gaussian noise and mixed α-stable noise, the LP-PAPGMC and LP-VKWPAPGMC algorithms exhibit superior performance compared to other affine projection algorithms.