This paper is an extended work based on the literature [fitted L1 method for delay fractional equations, (Cen and Vong in Comput Methods Appl Math, 2023)]. The highlight lies in the decomposition optimization of the solution for multi-singularity problems. Motivated by this idea, two corrected L-type schemes are constructed, of which optimal convergence order reaches 2 - α and 2, respectively, where α∈ (0 , 1 ) is fractional order. Significantly, the correction terms share the same forms with the discrete convolution structure for the Caputo derivative, which implies that the computation and analysis of these two parts can be integrated together.