Q: I have what looks like a loose pipe under my hood and I'd like to know where it goes so I can save some money and reconnect it

My boyfriend was driving one day I'm not entirely sure what happened but he said the car started smoking a lot and it had cut off on him. He had to wait for it to cool down before he could drive it again. After being parked for nearly a month I decided to drive it to work to see the problem myself, i only drove it a few blocks it didn't over heat but it did get hot fast past the middle of the temperature gauge. And I'd see puffs of smoke coming from the hood occasionally. I took a look under the hood and I noticed I think a pipe loose but I can't figure where it came from to reconnect it so I can see if that was the problem. the pipe is thin with foam connector on the end of it, it's curved around a box to the right of it between the box and the radiator. i don't know what the box is called, it's not the fuse box it's larger and is directly behind the left headlight.

My car has 11054 miles.
My car has an automatic transmission.