Pressing accelerator causes electrical short

Hello, so there is no 2.3-liter V6 Santa Fe, I think you mean 2.7Liter. And is this the GLS or LX model? I will try to help you out, but if you had a wire shorting out, it would most likely blow a fuse.
There is a 2.4-liter engine
Also, some more information would help, is this possibly a very fast clicking noise? And where exactly is the noise coming from?

Well, I don't think it has anything to do with injectors, how does the car run in general? Is it rough at idle? And for your noise, are you saying something like a zapping type noise?
So, it must be the 2.7-liter engine, that's the only GLS. Is this automatic transmission or standard shift?
And do you smell anything funny? Like electrical burn type smell? If there was an electrical short happening, you would smell it.
If you open the hood, do you hear anything in the engine compartment, with the vehicle in Neutral and parking brake on for safety. Have someone push on the throttle and see if you hear anything in the engine compartment.
Because we can only go by what you tell us. If possible, make a video with your phone or someway so we can hear the noise and upload it here.
And does this happen when the vehicle is sitting still, and you push on the throttle, or does it need to be moving to happen? Every little detail helps. Try to record the noise somehow as well. Video, whatever works for you.

If you did smell electrical burn, then yes that's a major concern. It's hard to say what it can be, cars have miles of wiring in them, are you able to take a drive with someone who can do a video so we can at least hear the sound.
The only real part of the electrical system behind the glove box area is the blower motor for the heating and A/C system. Does this vehicle have Automatic HVAC or Manual? Meaning can you set the temperatures different for the driver side and the passenger side?
The technical service bulletin (first pic) is for poor Ground wire issues that can cause the blower motor or headlights to have issues. But what you're describing sounds more like a power wire shorting to ground somewhere in a circuit before the fuse, that's the only way a short to ground would happen. Otherwise, a fuse would blow.

Yeah, it's just real tough to tell what's going on from my standpoint but having anything electrical burning is definitely not a safe thing. There are just too many possibilities of what it could be. I have seen vehicles with power steering fluid leak just catch on fire, even though it wasn't even running. It was leaking onto the exhaust and at our shop it was outside and happened just sitting there. It's not exactly the same as your issue but you get the point. Always better to be safe. If it's that dangerous, don't drive it. If you can't get a video, it's okay, just don't risk your own safety over it.

Okay, I will take a look at it and get back to you.

I don't see any recordings.

Okay, that to me sounds like a relay turning on and off really fast, that's not an electricity jumping to ground. What an automotive relay does is turns on a high current circuit using a low current circuit. They do this so that things like computers in the car can turn on Higher current circuits without damaging the computer. Relays over time develop carbon build up on the contacts and this causes them to not be able to keep their connections made. Thats what I am hearing, although I'm not in the car with you. So, what ever circuit this relay controls may not be shutting down because the circuit is not really being turned off, it's on again, off again, on again, off again, and happening so fast that the component it controls just isn't shutting down. For instance, if that was controlling your heater blower motor, it's just not fully losing power as an example. But it will need to be diagnosed since it's hard to tell which side its actually coming from. But I have heard them do that many times. It's a sign of it starting to fail. I would also have the charging system checked, specifically that the alternator is not putting out any AC voltage ripple and it's putting out the correct DC voltage. But take it to a technician you trust. Don't let them sell you a bunch of stuff you don't need, if you're in doubt of what they say, just ask us. We're always honest here. We're not selling anything. Just helping people. Here's a guide to relays except yours is under the dash somewhere. Most likely in the fuse panel. If you can access it while this is happening, you'll be able to feel the relay itself doing this. Added a picture of an automotive relay too.

Let me just add that if there is a loose ground wire for this relay, it will cause the same condition.

https://www.2carpros.com/articles/how-to-check-an-electrical-relay-and-wiring-control-circuit