Welcome to the forum!
I have a couple of questions, to make sure I understand what you want to do. Using your 2nd attachment, with the red oval, as an example.
Do you want the result to be oval shaped? The way I understand your message, the answer is no. You only want the part of the map which lies inside the red oval, right? So that the bottom part will be sort of oval shaped, but the top part will have the shape of the counties or districts, or whatever the smaller units are.
And the background map is vector, right. What format is it? SVG, EPS, AI, or other? I might need to see that vector file, but maybe not.
More than likely, when you click on any part of that map, the entire map is selected. Is that right? If that's the case, then this is the very easiest way to do it:
-- start with nothing selected, and then do Ctrl A (this will select everything - both the map and the red oval
-- Object menu > Clip > Set
There are a couple of reasons why that might not work for you. But since it's so easy, and it will take typing so much text to explain why, I suggest that you just try it. If you find some reason why it doesn't work, we can move on to more involved solutions.
For the one with the yellow, it looks like there are 2 ovals which are overlapping. Is that right?
It looks like the same country or whatever it is, as the one with the red oval, except that it doesn't have the white background. Also, it appears to have some other highlighting or something. And plus, wherever the white background is missing, the yellow oval is not showing either. Do you know why it looks like that?
The reason I'm asking about that, is because it looks like you might have used exactly what I suggested for the red oval, on the yellow one, except that you didn't use the yellow ovals for the clipping path. Instead it looks like the yellow ovals might be clipped with the path that goes around the whole country. If that's what happened, they you need to undo that.
If you don't know why it looks different from the one with the red oval, then....well you can probably use a very similar thing for that map. But the reason I say probably, is because parts of the yellow ovals are missing. So I'm not sure if they can be used for a clipping path (like with the red oval).
Would it be possible for you to share that file? I mean the vector file, with the yellow ovals. I'm a little hesitant to suggest the same thing for the yellow one, since parts of the yellow ovals are missing, and I don't know why (or how). If you have an SVG file, that would be better for me, because I can't open EPS files.