How much current does is take to melt a cable?

I assume a BS4343 type connector. The Earth pin would not be burnt (damaged) by a high Earth current under any normal circumstances. Damage is usually caused by plugging or unplugging under load. You have incorrectly applied the derating factor, only 3 cores will carry maximum current if the cable is 3 phase, the Earth being zero, and the neutral only the residual from the 3 phases caused by phase imbalance.

Now the cable overheat problem, the question is what was the load? As the Earth conductor is damaged I suggest that the problem was caused by a welder or similar load, being connected across the cable during use, perhaps welding structural steelwork connected firmly to Earth, although welders usually have completely isolated from Earth outputs, but perhaps some idiot has tried to connect the output work terminal to the welder Earth? The current in the Earth could easily be several hundred amps if this or some similar accident has happened, which would cause damage.

I assume a BS4343 type connector. The Earth pin would not be burnt (damaged) by a high Earth current under any normal circumstances. Damage is usually caused by plugging or unplugging under load. You have incorrectly applied the derating factor, only 3 cores will carry maximum current if the cable is 3 phase, the Earth being zero, and the neutral only the residual from the 3 phases caused by phase imbalance.

Now the cable overheat problem, the question is what was the load? As the Earth conductor is damaged I suggest that the problem was caused by a welder or similar load, being connected across the cable during use, perhaps welding structural steelwork connected firmly to Earth, although welders usually have completely isolated from Earth outputs, but perhaps some idiot has tried to connect the output work terminal to the welder Earth? The current in the Earth could easily be several hundred amps if this or some similar accident has happened, which would cause damage.

A good point (welders, unsuitable connection of leads, and resulting damaged earth conductors)!  Articles like IAEI and the alarming damage-pictures in EC&M show the risk of giving users the idea that the 'earth' clip should just connect to something apparently earthed.  In the US they've changed its name from ground to workpiece to try to encourage sensible use.