World’s first battery fuelled by air

I don’t think so. No more than your car is fueled by air when you have combustion. The difference is between carrying around oxygen in the cell, and drawing some oxygen in from the air in order to complete the reaction and release energy.

If you go to the group’s website, they explain the basic process:

On discharge, Li+ from the electrolyte and e- from the external circuit combine with O2 from the air, the process is reversible.

The big win is not carrying around Oxygen, which is more than twice as massive as Lithium. There’s also carbon and a catalyst involved, but of course a catalyst gets reused, and so you don’t need a stoichiometric fraction of that present. In the article at Green Car Congress it’s mentioned that the catalyst is Mn.

The Guardian article also claims

And as the cycle of air helps re-charge the battery as it is used, it has a greater storage capacity than other similar-sized cells and can emit power up to 10 times longer.

I see nothing supporting the claim that any kind of recharging is going on.