ABSTRACT In this paper, we present an updated version of our model (KYNXiltr) which considers thermal reverberation of a standard Novikov–Thorne accretion disc illuminated by an X-ray point-like source. Previously, the model considered only two cases of black hole spins, and assumed a colour correction factor fcol = 2.4. Now, we extend the model to any spin value and fcol. In addition, we consider two scenarios of powering the X-ray corona, either via accretion, or external to the accretion disc. We use KYNXiltr to fit the observed time lags obtained from intense monitoring of four local Seyfert galaxies (NGC 5548, NGC 4593, Mrk 817, and Fairall 9). We consider various combinations of black hole spin, colour correction, corona height, and fraction of accretion power transferred to the corona. The model fits well the overall time-lag spectrum in these sources (for a large parameter space). For NGC 4593 only, we detect a significant excess of delays in the U band. The contribution of the diffuse BLR emission in the time-lag spectrum of this source is significant. It is possible to reduce the large best-fitting parameter space by combining the results with additional information, such as the observed Eddington ratio and average X-ray luminosity. We also provide an update to the analytic expression provided by Kammoun et al., for an X-ray source that is not powered by the accretion process, which can be used for any value of fcol, and for two values of the black hole spin (0 and 0.998).