Push and current
Wrong Track: More current flows out of the extra battery.
Right Lines: Extra batteries provide a bigger push on all of the charged particles in the circuit. When a second battery is added to the circuit, the positive side of the combined battery becomes more positive with respect to the negative side. A bigger force is therefore acting on all of the charged particles in the circuit, as they are pushed away from one side of the battery and attracted towards the other. Adding a battery, therefore, results in the same charged particles (in battery, bulb and connecting wires) moving around the circuit more quickly. More charged particles pass each point per second. In other words, the current increases.
Charged particles don't come from the battery
Thinking about the teaching
This incorrect line of reasoning is rooted in the idea that the battery is the source of the charged particles which make up the electric current.
Here you should emphasise that:
Teacher: The charged particles don't come from the battery. They start off in the battery, connecting wires and bulb. The bigger current is due to those charged particles moving around more quickly.
Expressed in terms of the rope loop:
Teacher: It's like with the rope loop. The rope didn't come from me in the first place. The loop is already there and, as when a second battery is added, I simply moved the same loop around more quickly.
Thinking about the learning
When moving from one to two batteries in a circuit, pupils often anticipate (sensibly) that the current will double in strength. What happens in practice is that the current certainly increases, but not to the extent of doubling. The reason for this is that as the current in the bulb increases, the filament heats up more and its resistance increases. The increased thermal agitation of the atoms of the filament makes it harder for the charged particles to pass.