How to stop Illustrator drawing over holes

I have a template in illustrator that has (as far as I can tell) circular holes in it. When I select the whole shape, I get the blue outline line round the edge but also inside the holes - so I am assuming that they are holes rather than drawn-on circles (if that makes sense).

What I want to do is make coloured rings around these holes. I had assumed that if I simply draw a circle that is larger than the hole, it will not draw onto the hole and instead leave a ring around it. The problem is, it simply draws a circle covering the hole. How can I get Illustrator to treat the hole as a hole, and not draw over it?