The onset of electroconvection in binary mixtures of a bent-core and a rodlike nematic has been characterized by measuring the threshold voltage U(c) and the critical wave number of the pattern in a wide range of frequencies f. In the mixtures rich in bent-core molecules, a "conductive-prewavy2-patternless-prewavy1" morphological sequence has been detected with an unusual negative slope of U(c)(f) at high frequencies. This latter scenario seems to be related to the bent-core component, as it disappears with increasing the concentration of rodlike molecules. In addition, one of the parameters most relevant for electroconvection, the electrical conductivity, has also been varied by ionic salt doping. It has been found that the above effect of the banana-shaped molecules on the electroconvection scenarios can be suppressed by the conductivity.