Redex is a fuel additive, containing detergents that will help keep your injectors and combustion chamber clean by eliminating Carbon deposits.
There are various other equally-good fuel additives on the market, including Millers, Forte, BG44K, and others.
Alternatively, you can use premium fuels such as Shell V-Power or BT Ultimate etc, these fuels already contain the enhanced additives pack, and it's easier to just fill-up at the pump and not worry about adding additives from a bottle, though use of premium fuels will be overall more expensive than adding the additives yourself.
That said, fuel additives won't fix a faulty fuel pump (or any other mechanical part in the fuel delivery system).
On modern Diesel engines, the high-pressure fuel pump is being lubricated by the Diesel oil as it is flowing through it, and running the pump dry (and unlubricated) can indeed cause damage. If this happened, then the only remedy is a new (or refurbished) pump - no fuel additive will help. However, this sort of damage usually occurs when filling-up with petrol instead of Diesel oil, causing the pump to work for a period of time with poor lubrication. Simply running out of fuel shouldn't damage the pump in this way, because when the engine cuts-out you'd stop the car, which will in turn stop the pump. So it really depends on how long the fuel pump was working without proper lubrication. What you would normally get when a Diesel engine is allowed to run dry, is air entering the fuel system, which will in turn need to be bled, but this in itself should not cause any permanent damage.
On petrol engines, running the engine dry shouldn't have any consequences and everything should be back to normal once you put some fuel in it.