The item your looking at is an EGR valve. The ufo looking thing. The bottom of the egr valve is supposed to be connected to that open pipe. Something will need to be done, because that's going to have exhaust gases coming out of it. Your EGR recirculates usefull exhaust gases back into the intake to reuse, to save gas and reduce emissions and it actually cools the engine temps(DING DING DING)(With hot exhaust gases leaking in your engine bay and no NOX exhaust gases to cool the engine, it may be worsening your problem)
Fixes
1. Re hook it up. But if it was unhooked, the egr may have been bad. Replace EGR valve if bad.
2. Follow EGR delete(Found in HowTo section or on Focushacks.com)Will throw a code.
3. Remove all EGR components, including DPFE, Block of EGR pipes on Intake and Exhaust manifolds. Will throw code.
Both 2 and 3 you could get a tune to delete the egr so you don't get a code.
Also....The leaking thermostat housing is a common problem on the zetec engine.
It's caused by a bad factory valve cover gasket causing dripping oil into the crack of the housing and destroying the gasket and plastic housing. You should replace the valve cover gasket and then the thermostat housing w/ (A new gasket, Yes there should be one between the block and the housing)