hi,
I am new to the forum and currently own an Audi A3 (1.6L), 8L. I have two issues and have been scratching my head to get over them. One there is a black wire with plug dangling on top right side of engine bay pictured below

the dangling wire
I have no idea where it goes, have tried taking to mechanic but after two hours of searching, he had no clue.

Secondly, there is one pipe going to the radiator on top pictured below which gets hot when engine is running

and one pipe coming from the radiator below which is never hot. Similar to previous issue, have spent considerable amount on mechanic and the fault became worse instead of getting better after replacing thermostat. My car used to get hot when i drove on motorway but after the visit to mechanic it just simply gets hot in ten minutes.
Any suggestion, help or advice is welcome. I am an international student and spending everyday on an old beater car for a "maybe" solution is not an option