The nature of the progenitors of Type Ia supernovae (SNe Ia) remains a mystery. Comparing theoretical rates and delay-time distributions of SNe Ia with those inferred observationally can constrain their progenitor models. In this work, taking thermohaline mixing into account in the helium-enriched main-sequence (HEMS) donor scenario, we address rates and delay times of SNe Ia in this channel by combining the results of self-consistent binary evolution calculations with population synthesis models. We find that the Galactic SN Ia rate from the HEMS donor scenario is around 0.6-1.2 x 10(-3) yr(-1), which is about 30 per cent of the observed rate. Delay times of SNe Ia in this scenario cover a wide range of 0.1-1.0 Gyr. We also present the pre-explosion properties of companion stars in the HEMS donor scenario, which will be helpful for placing constraints on SN Ia progenitors through analysing their pre-explosion images.