Can these connectors cause the rich codes if they are not securely connected? If so, what is the name of that hose?

 

Generally they cause vacuum leaks and yes they have caused problems. It's part of the PCV system and you also have a black round disc on the passenger side of the engine that can also leak. They have an O-Ring on the hose that you can replace.
.
.
.

 

If you're getting P0400 pay attention to what Jaguar say in workshop manual.

It might cause knock-on codes.