The noise you're experiencing are compression artifacts. As you get under about 25Mbps or so bandwidth and below you get less and less latitude in your ability to do color correction. As you get under about 7.5Mbps, the artifacts will start becoming more apparent without doing anything to the video [I'm using 1080 as a baseline.] The more bandwidth you have, and the more bit depth you have, the more the video can tolerate manipulation without introducing too much (noticeable) noise.
Karsten's suggestion is one way. FCPX has changed if you haven't noticed, and there is another (new) way — but very similar!
Colorizing should work somewhat better than color correction (in keeping the introduction of artifacts down to a minimum.)
Add a Color > Colorize effect to your clip. Mouse over the name/titlebar - you'll see a mask icon appear.
Clicking on the mask icon, you have two options: Shape Masks and (1) Color Mask (you can add as many Shape masks as you need, right with this effect! Also notice you can *invert* the masks.) Select the Color Mask and click and drag across the white napkin (keep the expanding circle inside the bounds of the napkin). If you create holes in the rest of your clip, don't worry about that. Just select an additional Shape mask and draw an outline somewhat closely around the edges of the napkin. Click on the Shape Mask "layer" in the Effect inspector and drag it to under the Color Mask. Then click on "Add" and from the dropdown and change it to "Intersect". That should eliminate any holes you might have created with the color mask.
To get the best result with the color mask, hold down the shift key and keep clicking obvious pixels that aren't included to add them to the total mask. To take care of any aliasing you cannot get rid of, increase the Softening slider value.
If the scene moves, you don't have to worry about the color mask, but you will have to keyframe the shape mask to "track" the masked region. You can make changes to location, size, feathering, and rotation as you track. If you've never tracked a region of video, set the Playhead within the clip (basically anywhere - you can track forwards and backwards) and click the diamond shaped icon to the right side of the shape mask's icon*. The easiest thing to do is to hold the shift key down and type the forward or backward arrow key a couple of times. Each shift - arrow tap advances (or retreats) the playhead 10 frames. If the motion is smooth, you can go further before moving and positioning the mask at the new position. Play through the section to see if the shape tracks correctly. If it doesn't, try to find the middle point of change and add move/reposition the shape. Work down to a frame by frame if necessary (it usually is not and the wider gaps are usually sufficient.) If you do a really good job, it is quite worth the effort when you get what you want 😉
*Normally, you do not have to re-set a keyframe - FCPX sets them automatically at any change in the object AFTER the first keyframe is manually set. You will need to set manual keyframes if the motion stops for a pause, then resumes again. Manually set a keyframe where the motion stops then another where it resumes.
You would have basically the same problems with a Key. The benefit of not using a key is that you don't have to worry about multiple layers of clips/backgrounds. An alternative effect to use is the Hue/Saturation effect. Turn down the Value parameter for black.
HTH
