Is there a way using Windows API calls to simulate a mouse click at a
specific location WITHOUT moving the mouse to that location? Most
procedures using SetCursorPos, which is what I want to avoid.

If you know the handle of the control that you want to click (assuming
it has one) then you could uses SendMessage with WM_LBUTTONDOWN then
WM_LBUTTONUP.

You could look at the SendInput API

Personally I would move the mouse, do Mouse_Event and then move the
mouse back to its original position.

The control does not have a handle. Also, I do not want to move the
mouse at all. Any ideas?

Strictly speaking if it does not have a Handle then it is not a Control
- it will be a pseudo control drawn on a real one

Did you look into the SendInput API ?
- from MSDN I could not figure out whether a Click paid any attention
to the mouse co-ordinates in the Structure.

Why don't you want to move the Mouse ?
Maybe there is a workaround for the reason why you don't want to move it.

You might be better off asking in a programming News Group.