We report on a theoretical work on magnetotransport under terahertz radiation with high-mobility two-dimensional electron systems focussing on the radiation power and temperature dependence. On the one hand, we study the interaction between the obtained radiation-induced magnetoresistance oscillations (RIRO) and the Shubnikov-de Haas oscillations (SdHO) from the power dependence standpoint. We obtain strong modulation of the SdHO at sufficient terahertz radiation power. On the other hand from the temperature dependence standpoint we obtain an important result: the range of temperature where RIRO are observed can be largely extended by using terahertz radiation. In the terahertz region we still obtain RIRO up to 20K at a radiation frequency of 400GHz. Since an increasing T gives rise in turn to more disorder in the sample, we would expect also the observation of RIRO with terahertz radiation when using low-mobility (high-disorder) samples at low T. This work is supported by the MINECO (Spain) under grant MAT2014-58241-P and ITN Grant 234970 (EU), Grupo de Matematicas Aplicadas a la Materia Condensada, (UC3M), Unidad Asociada al CSIC.