An incremental least-mean M-estimate algorithm is proposed for distributed network, which minimises a robust M-estimator-based cost function against impulsive noise via the gradient descent method. Moreover, based on the same frame for implementation, an incremental distributed least-mean p-power algorithm is presented for combating impulsive noise. Simulations verify the superiority of the proposed algorithms in the presence of impulsive noise relative to some existing incremental distributed algorithms.