Current-time responses having a singularity at time t = 0 are typical of some transient electroanalytical experiments (such as e.g. the potential step chronoamperometry). A reliable simulation of such singular responses cannot be performed by conventional simulation techniques, as they fail to provide accurate results close to the singularity. A recent extension [L. K. Bieniasz, J. Comput. Appl. Math. 323 (2017) 136] of the adaptive Huber method for solving Volterra integral equations (IEs) is adopted for the simulation of singular responses at planar and spherical electrodes, in cases when one-dimensional, semi-infinite diffusion transport is coupled with (pseudo-) first order homogeneous reactions. Highly accurate approximants are elaborated, for computing certain integrals of the kernel functions occurring in the related IEs, and needed in this simulation approach. Formerly published approximants to the coefficients of the adaptive Huber method are also improved. The resulting algorithm is tested on four examples of IEs describing chronoamperometry for ErevCirr, ECrevE, CrevE and ErevC'(irr) reaction mechanisms. Singular transients are simulated automatically with a prescribed accuracy, also for t arbitrarily close to the singularity. (C) 2018 Elsevier Ltd. All rights reserved.