When I started my car, the low coolant light went on and I found that the expansion tank was dry. I also noticed that the narrow hose leading into the top of the expansion tank makes a T-junction with other hoses not too far in front of the tank, and the lower hose at the T-junction had come loose. When I tried to reconnect the hose and tighten the clamp, I lost the clamp in the engine somewhere. I noticed some "gook" where the hoses connect so I am wondering:do I just need to buy a new clamp and reconnect the hose or is there some sealant I need to put at the connection (the gook, perhaps), or is this perhaps a more serious problem. Thanks.

 

Theres no sealant used on those hoses. Is the gook maybe some sort of sealant that someone used in a previous repair attemt. I would buy a new clamp and if it still leaks replace the t fitting and clamps.

 

I have an 2000 a4 1.8t I was wondering what the hoses on the back of the motor are it has an electrical connector on it?

 

Sounds like your upper hose is leaking which isnt good. There is a o-ring that seals the disconnect that is leaking on my car and making a smell when it burns off.
So if I were you I would just replace the hose. The rubber gets old especially on the upper since this is the hose that has the hot side of the engine is going
I have a 2002 Audi A6 2.7T and this link is the cheapest I can get it.
I hope this helps you.
OES Genuine W0133-1736438 Radiator Hose | Auto Parts Warehouse