Can you remove the section where the line goes through the hole, I do not know what it would be called.
If you're talking about the first picture, it's called a "fairlead". You can remove it, but you need it to guide the line into the cam cleat... therefore it's not a good idea unless you replace its purpose with another type of fairlead. There are many types of fairleads... for instance in the picture I posted, the block acts as the fairlead. There are cam and CLAM (2 different things) cleats that have the fairlead integrated into the assembly. The metal cam cleat you picture actually has an eye strap straddling the cams that could be called a fairlead, but its purpose is to keep the line captive, rather than guide it into the jaws of the cleat. On a swiveling device you especially need the fairlead...or you will quickly foul the line and probably end up getting wet!
Okay, Joe, let me give you a huge piece of advice, if you don't mind, of course.
Go to any sailboat hardware vendor, or any good online chandlery, and study all the pages of hardware they have. There will be illustrations galore. Then go into a real store and hold them in your hand to see how they work. That's what every sailor in this forum has done when they first started. It's part of the learning process, like learning sailing and boat nomenclature. You should also have started a library of sailing and boat information and instruction. The best is "The Annapolis Book of seamanship".... you can find it in the library, and older editions at any used bookstore. Good Luck, Have Fun.