Gas Gauge not working .?

It is an electrical issue and the shop is trying to shaft you.

. go to another shop.  Even the dealer would be cheaper than this dude.  I mean that is my guess without seeing a wiring diagram. So IMO  Why they would tie the pump to the fuel level sensor is not sensible to me(but they are both tied to the key switch...so the problem could be there)[I am thinking aloud]   Electrical gremlins are the worst. You need testing equipment so you can rule out working parts.  Electrical components is not something you can TAKE BACK.  Once you bought it, you own it. That could get costly WITHOUT FIXING IT.  It could be a ground wire, though I am not certain where that is located.

. With your choices that you have discovered, I would have gone with the sending unit. That relay should be in the fuse box.  The actual float is in the tank.  Those with the wiring schematics can see that, which I can not.

The fuel pump and fuel level sensor could be combined into a single unit. Try another shop and see what they estimate.

"Since your fuel tank isn't getting empty, it's gotta be the fuel pump. Without out it, the tank cannot empty." Sounds logical right? If you believe that, you will be out lots of money and nothing anybody can do about that. Take it to a reputable repair shop to find the problem and explain the electrical process. It sounds like a fuse could be the problem but you need someone knowledgeable in automotive circuitry and a diagram to show you the problem. If I had the diagram, I could tell you if it makes sense. If the gauge is not indicating properly, it could be a fuse. Usually, more than one item operates off of a fuse so something else may not work. If everything else is working, then it probably is the sending unit (the device connected to the tank that sends the signal to the gauge). If the Hyundai has an integral fuel pump/sending unit, then it may indeed require a new pump. Without a circuit diagram, it is a guessing game.

It just might be a bad ground connection.

It's possible.  The fault was either with the gauge, its wiring or the sender unit. I presume the shop has ruled out the first two. In some cases these days the sender unit is part of the pump, so that may indeed be what it needed.

The fuel gauge sending unit is part of the fuel pump assembly.

Not a fuel pump but a fuel sending unit which is the gadget attached to the pump assembly that sends a signal to the gas gauge on the dashboard. Some cars you are stuck replacing the whole assembly (pump and sending unit) which can be $600 just for the part. Could be a lose wire or corrosion interrupting the electrical signal to the gauge on the dashboard or the gauge itself that is defective.

Tell the shop to go ahead but ask them first if they will EAT the cost of the fuel pump if it does not fix the problem.  If they say no, go to another shop that will determine with 100 percent certainty that the fuel pump is really the problem.  It just might be a bad ground connection.

maybe you should get a mechanic to check it out

No, take it to another shop