Non-uniform height semiconductor quantum rings are studied in order to determine their electronic and optical absorption properties. Theoretical modelling of the structure includes an analytical description of the non-regular multi-hilled confining potential as well as the presence of repulsive scattering centre and external crossing electric and magnetic fields. We have discussed the features of localised and extended (rotational, Aharonov–Bohm-like) states in the presence of the magnetic field. A modification of the spectrum, with the appearance of a Stark-like behaviour, and its corresponding modification related to the repulsive potential is analysed when the electric field effect is considered. In double-hilled structures, these properties of the energy spectrum are of main importance in explaining the apparent optical transparency induced within a certain range of the electric field strength. The presence of the repulsive centre is found to cause a moderate redshift of the light absorption response. © 2020 Informa UK Limited, trading as Taylor & Francis Group.