2013 Kia Optima. Fuel gauge shows 1/16th of a tank and the low fuel light is not on. Engine shut down. Out of gas. Add gas. Car starts up. What would cause the gauge to misread in this way?

put jumper cables to the car like if you are going to jump start the car and then you might be able to open the door to the gas tank.

try using a fuel system cleaner. fill tank with gas and add to full tank. run car for an hour on freeway (roundtrip)....it should run
smoother. some gas has water in it and a fuel system cleaner should re-dissolve the water into the gas. sounds like a fuel-dlivery problem, dirty fuel filter, or moisture in the fuel/fuel-line.

If your model has a low fuel warning light, it would be a symbol of a gas pump and be near the gas gauge. The light is connected to the fuel gauge sending unit in the tank.

If the engine cranks the starter is doing its job.
You would need to check for fuel pressure at the injectors.
Its possible the pump was clogged up or overheated when you were driving it low on fuel. The gasoline keeps the pump cool and lubed.
The only fuel filter for your model is inside the gas tank.

This problem is caused by in most cases the fuel pressure bleeding down when the engine is shut off, this leads to air in the fuel line because the fuel leaks back to the pump and then into the fuel tank. The fuel pump is electric and it has a built in fuel pressure check valve in the pressure outlet of the pump. To confirm this is the problem you must have a fuel pressure leak down test run, the test port to check fuel pressure is on the top of the engine in the pipe that supplies fuel and sets over the top of the fuel injectors. You can purchase a fuel pressure test gauge from most auto parts stores for about $20.00. Hook the test gauge up to the test port (it looks like an oversized tire valve) start the engine and verify you have no leaks at the connection of the gauge, once that is sealed and has no seepage of fuel note the fuel pressure at idle, shut the engine down and watch the gauge, it should not drop more than 4 pounds in one hour, in most cases it will drop off right away. If the fuel pump is the problem you will need to replace it, to do that requires you drop the fuel tank down out of the car, this is a very difficult job on the ground, you will need to raise the car as high off the ground as possible, support it on jack stands and then with a hydraulic floor jack support the fuel tank and remove the filler neck then remove the two steel straps that secure it to the cars underside, once the straps are off slowly lower the tank and remove the fuel lines and the electrical connector from the fuel pump/sending unit connector, then remove the tank and remove the retaining ring that holds the fuel pump/sender module inside the tank. Reverse to assemble, then put at least 4 gallons of gas in the tank to prime the pump.

check gauges means that the car senses something is not working properly, such as temp gauge reading to hot ,or oil pressure reading to low ,fuel to low, its to get your attention that something is wrong .

It is in the fuel tank....here is the directions on how to do the job:

Removal & Installation

1. Before servicing the vehicle, refer to the Precautions Section.
   
2. Relieve the fuel system pressure.
   
3. Disconnect the negative battery cable.
   
4. Remove the rear seat cushion from the vehicle.
   
5. Clean any dirt that has accumulated around the fuel pump cover so it will not enter the tank during pump removal and installation.
   
6. Remove the fuel pump cover.
   
7. Disconnect the fuel gauge connector, hoses, and the gauge.
   
8. Disconnect the fuel pump electrical connector.
   
9. Remove the fuel pump from the bracket assembly.
   
10. Remove the seal ring and discard.
    

To install:

1. Clean the fuel pump mounting flange, fuel tank mounting surface and seal ring groove.
   
2. Apply a light coating of grease on a new seal ring to hold it in place during assembly.
   
3. Install the seal ring.
   
4. Install the fuel pump to the bracket assembly carefully to ensure the filter is not damaged. Be sure the seal ring remains in the groove.
   
5. Hold the pump assembly in place, and pull the fuel pump down so that it is tight against the bracket.
   
6. Connect the fuel pump electrical connector.
   
7. Connect the fuel gauge, hoses, and gauge connector.
   
8. Install the fuel pump cover.
   
9. Install the rear seat cushion.
   
10. Connect the negative battery cable.
    
11. Start the engine and check for proper system operation and for fuel leaks.
    

P0442 EVAP Leak Monitor .040 Leak Detected (Medium Leak)

P0456 EVAP Leak Monitor .020 Leak Detected

replace your gas cap and clear codes, then wait and see if light comes back on, if it does you have a more serious problem like a fuel tank leak or a defective fuel tank pressure sensor.

I had the same problem along with a rough idle and found out it was my EVAP canister.