We have used mean-field theory based on the Bogoliubov inequality for the free energy to study the effects of random transverse single-ion anisotropies and magnetic field on the mixed spin-1 and spin-1/2 Blume–Capel quantum model with the coordination number z = 3. The interactions of the transverse crystal fields Dx and Dy act only on the spin-1 sites and are randomly active with probability p and q and inactive with probability 1−p and 1−,q respectively. The thermal behaviours of the order parameters are studied to determine the nature of phase transitions and to calculate the phase diagrams on the ($\varphi$x = Dx/Jz, k$_B$T/J ), (p, k$_B$T/J) and (q, k$_B$T/J) planes. It is found that the model exhibits only second-order phase transitions. The compensation temperatures are also observed and their lines, T$_{comp}$-lines, are depicted on the ($\varphi$x ,k$_B$T/J) planes. The hysteresis loops are obtained by introducing an external magnetic field on the system which reveals that the coercive field decreases with temperature and with positive values of $\varphi$x and $\varphi$y . It is also found that remanent magnetisation increases with negative values of $\varphi$x and $\varphi$y.