When current flows, Is there any potential difference?
Entering into the world of super-conductivity things get decidedly odd.

In your case, with a superconducting wire across the battery, as you envision, measuring the voltage across any two points will yield a zero value since the resistance is also zero. This however, does not preclude the passing of current. Just as if it were a zero ohm jumper in a conventional circuit, you would not be able to measure any volts across it even when you know there is a current flowing.

In theory, with an ideal battery, the current would be infinite in the circuit you showed in order to obtain the battery voltage. However, in reality the battery has some resistance so you would be limited to that amount of current.

Of course, getting the battery to work at superconducting temperatures would be a trick all on it's own.