Good morning,
Having a bad injector would not present much of an issue when starting, it would however cause a miss in the engine at all times while its running. You would surely notice it. And the computers would notice it and throw you codes ranging from a cylinder misfire to emissions-related codes for being too rich.
There are bunch of things that could cause what you are experiencing, for instance, a sensor, pressure regulator, low fuel pressure, fuel leak just to name a few.
That being said, Do you have a check engine light on on at all?
Do you have a way to measure your fuel pressure?
Your vehicle has two fuel pumps on it by the way. There is the one that lives in the tank (referred to as a lift pump) and its job is to pump fuel from the tank up to the other fuel pump called the high-pressure fuel pump (located on the engine) so it can then feed the fuel injectors with the high pressure they need).
Before we get off the rails here though, please let me know if you have a check engine light on.
Jon
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Tuesday, December 15th, 2020 AT 11:30 AM