Sparse and low-rank tensor recovery has emerged as a significant area of research with applications in many fields such as computer vision. However, minimizing the \(\ell _0\) -norm of a vector or the rank of a matrix is NP-hard. Instead, their convex relaxed versions are typically adopted in practice due to the computational efficiency, e.g., log-sum penalty. In this work, we propose novel log-sum regularized Kaczmarz algorithms for recovering high-order tensors with either sparse or low-rank structures. We present block variants along with convergence analysis of the proposed algorithms. Numerical experiments on synthetic and real-world datasets demonstrate the effectiveness of the proposed methods.