I'l assume you are talking about the High Pressure Fuel Pump (HPFP) in a diesel.
If the pump is faulty it could damage an injector, but the injector itself should be tested separately, as the
problem could be from the pump not supplying enough pressure to the rail, or the rail. If there is no pressure from the HPFP, then this should be registered by the rail sensors, and most probably all injectors will fail. There are stands for testing injectors, they produce the exact pressure of your pump. If there is no damage, then the obvious
problem is the fuel supply pressure or lack of it. The injector should be always tested. If there is no stand nearby, this is done by switching the injectors, from one cylinder to another or replacing it with one from the
shop. If the
problem remains in the same cylinder then it's not the injector. If the
problem goes to the other cylinder then most probably is the injector. After that there should be an inspection of the cylinder and other components, with the small cameras, as there could be significant damage from the faulty fuel spray or metal debris .
You should check what of those things have been done when they diagnosed a faulty injector. And if you have your old injector go tests it separately. If there is nothing wrong with it, then there is a big
problem with that garage and you could have a very big claim, at least a full refund of the injector. But if the injector is bad, then..... bad luck.
If the pump is faulty, then there are two major things, that could have happened. The valve that sits on top, which is pretty good as there will be no damage from metal particulates(just some electrical issues), the other one is a mechanical failure and metal bits going around. In both cases having only one injector and a HPFP failed is a very rare case. It will most probably affect not just one injector. If there are metal debris than the rail itself, or the valves or cylinder. Imagine metal particulates going around with fuel with enormous pressure. If that happens most probably you will not start or have a very very bad sound coming out.
Other faulty thing could be some gasket problems, or something clogging up the fuel free flow. This results in low pressure and lack of fuel, that should appear somewhere along the pressure sensors.
Also you should check the fuel supply pipes. If there is something wrong with them (like blocked from bad fuel), they could produce lack of pressure too.
Now engine cutting of around 2000rpm, could be from lack of fuel, fuel pressure, exhaust problems, EGR problems, incoming air, even high amount of water in the fuel, etc. This means that there is a reasonable amount/quality of fuel,pressure,air,exhaust until that point, but after that something goes wrong. The fault code should point to something.
Good Luck.