My injection light come on after a few minutes of driving. What does this usually mean?
Are there any "first-check-this" tips?

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Basically there's no other reliable way to diagnose this than to hook up the scanner and read the DTC-s.
This fault can be triggered by many things, including 8 engine management sensors, timing variator solenoid, fuel injectors etc.

https://www.fiatforum.com/barchetta/485767-injector-dash-light-2.html?p=4589690

Take a look here, just had the same thing, only after about 5km driving at highway speed, on the end it was lambda probe replacement, but I agree, this can be all sorts, buy the cable from the chap o mentioned, excellent seller, good luck, keep is posted