Negative battery cable hot/car not starting?

Hello, the battery cables need to be tight and secure. If the cable is getting hot and smoking, there is too much resistance, most likely between the battery post and negative cable itself. The proper way to check the connection is to check for voltage drop from the negative battery post to the engine block with the key on, and engine running for a minute if it will start, if not, then just key On, and turn the headlights on as well to load the electrical system some. set your multimeter to DC volts and make sure the meter lead is on the battery post, not the cable, you need to see the voltage drop that's occurring between the battery post and cable and put the other meter lead touching the engine block. So, it's a ground side voltage drop test. If the cable end is broken, it needs to be replaced. A bad ground like that can damage other components. The starter is clicking because there isn't enough current flow due to the high resistance at the broken negative cable end. Can you take a picture of the negative cable so we can see what the broken end looks like?
The starter motor pulls hundreds of amps, that's why the cable is getting hot.

https://www.2carpros.com/articles/how-to-use-a-voltmeter

You want the voltage drop from battery negative to the engine block to be close to 0volts, Check from battery negative to the vehicles body(chassis) as well.
You can do the same test on the positive side as well, from battery + to the positive wire on the Starter motor. Point to point testing, should read close to 0volts as well.

Well, wait until you get the new cable on there before you go hooking up any jump boxes or anything, Try turning the engine over by hand if you suspect a mechanical issue. Just use a socket on the crankshaft pulley and a large bar, you should be able to turn it over. You'll be pushing against the compression of the cylinders, so it will be a little tough, but it should rotate, you can also pull the spark plugs and then it will turn over by hand really easy.

Leave that jumper box off until you get the battery cables all sorted out. With the negative cable off you don't have a complete circuit right now.

You're not able to turn the engine over by hand?

That's not good, are you thinking the starter gear may be engaged into the flex plate? And I assume it didn't start at all after the new battery was installed and the cable was overheating?

You might try taking off the main drive belt as well and see if it will turn over, just to be sure it's not a seized-up alternator or ac compressor. Thats easier than pulling the starter first, just to verify it's not something else. I'm wondering if this has some sort of vacuum pump or high-pressure fuel pump that's driven by a camshaft, when those seize up, it can cause issues like this as well, I'm just looking at some diagrams of the engine here. Do you see steel fuel lines going to the fuel injectors?

The problem with the high-pressure fuel pump is its mechanically driven by the cam, and when they fail, there ends up being metal shavings stuck in the fuel rail and into the fuel injectors. It can be a real mess. You have to release the high-pressure fuel in the rail before taking any of the steel lines apart, and they are usually a one-time use. The high-pressure pump can run 500 to 2000 psi, so you can't just unbolt the lines to it. I would hook up a scan tool and see what the fuel rail pressure sensor is reading for pressure once you get the battery cable fixed. Do you remember what the codes were?

Well it's difficult to say what happened at this point, you did have it running, and the HP pump has to be reinstalled at a certain position and service info here at a quick run through is stating the crank needs to be rotated until the cam follower is at its lowest point, And then install the cam follower into the vacuum pump and then the high pressure pump into the vacuum pump. So, this is no easy task to take apart, just to verify the engine will rotate.
I'd say pulling the starter is the easier thing to do, I take it this was an auction vehicle.
This looks like a serious job to replace this pump, some of this service info is talking about removing and installing a camshaft adjustment valve, there's pages here for just the HP pump. This requires special tools listed here as well.
I do hope you find the starter jammed up into the flex plate, when you get the battery cable hooked up, go in and check the live engine data PIDs and see if there's any fuel pressure reading. I'm trying to find some info on the starter location, there's quite a few design variations listed. ( 2.0L FSI manual, TFSI manual, TDI manual trans, ) Those are the 2.0liter listings.

The P0130 and P0135 are the B1S1 oxygen sensor (circuit) and (heater) faults, the other code is the intake manifold runner position circuit perform.

Looks like the Starter is under the air filter housing.