What will happen when I connect two batteries of different potential in a circuit like below? And if there will be a current will it be the same in the whole circuit? How is it possible since electrons can't travel through batteries (electrolyte) and cathode is depleted of electrons?
In the circuit you have shown the "batteries" are not in parallel, they are connected in series and the resistor is connected in series as well. By definition the current is exactly the same through elements connected in series, so yes, the same current flows through the resistor and the batteries.
Whether electrons actually travel through a battery is not important from a circuit perspective...the important things is that the current into one terminal is exactly equal to the current out of the other terminal of a given battery.
Current travels into the positive terminal of a battery whenever you charge a (rechargeable) battery. There is no inherent problem with a circuit that has current flowing into and out of batteries at the same time.
If your circuit has ideal voltage sources then their voltage will never change. If you imagine that the voltage sources are rechargeable batteries, then one battery will discharge and the other will be charged. When they reach the same voltage the current will stop flowing. As @james-large said, you drew ideal voltage sources and the situation changes if you think they are real, non-ideal batteries.