x-ray radiation is high frequency and typical electronic circuits (such as would include a resistor) are low frequency, so there is little direct effect between the electric field in the x-ray beam and the the circuit's current. However, there may be indirect effects. For example, the x-ray could heat up a resistor so that it's resistance increases. Or the x-rays could remove electrons such as in the photoelectric effect. But I think such effects are small enough that you will not see them unless you are looking for them.
If the radiation is at comparable frequencies (kHz-GHz) to those used in AC circuits, then the radiation will induce additional currents in your circuit, known as signal interference. For instance, when you turn on a blender, it's motor generates radio waves that make an analog antenna TV (remember those) show a fuzzy picture. Unwanted interference can be minimized by electromagnetic shielding. If the induced current is desired, then you get into the complex world of antenna design.