The Cursor in the middle is, I assume a cross-hair icon? If so, don’t make that the cursor icon but just a fixture in your UI that’s always in the middle.
Then hide the actual cursor when ever the game gets the focus, by registering to the Application.focusChanged event and in your callback receiver, make the cursor invisible and ideally also lock it to the center of the screen (which does set it as invisible). See here for code examples.
Locking the cursor means the position won’t change, so if your scripts rely on that to turn the camera, you need to switch them to react to the delta movement of the cursor instead.
I don’t get what you mean since the cursor is technically always in the middle.
Have you tried:
Cursor.visible = true;