Hello!
Okay, I'm assuming that the wave seal is no longer sealing properly due to the game, causing the pump to draw in air. This, in turn, prevents the injection timing from being regulated correctly because the pressure inside the pump is dropping. If that's the case, the error code in the fault memory should read 'Injection Timing Control Difference.'
for example, this plate with the 4 raised areas
I assume you're referring to the impeller; if this is worn out (due to fuel with poor lubrication, such as kerosene), the flow rate and thus the amount of fluid pumped will decrease; furthermore, the return springs could also be broken (this should be visible when the pump is disassembled).
Therefore, the pump CAN deliver more fuel if it wants to...
Okay - the engine control unit receives the deviation of the fuel quantity control valve, but not the actual amount of fuel injected

. In tuning, the fuel quantity control valve is deflected further, and thus more fuel is injected. However, if the pump delivers less fuel globally (i.e., across all speed and load ranges), the target amount will not be injected here either;
Who has enough knowledge about the pump's structure to say more about it?
What exactly do you want to know?
And isn't there a device that allows you to connect a sensor instead of a nozzle, which measures the pressure and amount of liquid dispensed by the pump? Or how else can I check the performance of the pump without having to send it to Bosch for 600 euros?
Yes, there is such a device, it's called a fuel injection pump test bench; it costs around XXXXXX Euros, and that's why the Bosch dealer also charges that much to check it; he has to recoup his investment

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Possibly find a used pump at a low price and install it

.
It was only during the installation that it was noticed that the axle had a relatively large amount of play forwards and backwards.
I would probably do something about that soon, because a slipping pump pulley can cause the timing belt to come off

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Best regards, Nebelwerfer.