I'm not sure of your experience, so humor me.
Generally with Illustrator, your options are: delete/remove vector structures, clipping masks, or opacity masks.
I think you need a clipping mask. A compound clipping mask, to be precise.
If you've never made a clipping mask, do a test.
Draw a big circle, then draw a small square on top of the circle and position it so it overlaps the outer edge of the circle.
Then select both objects and Object > Clipping Mask > Make.
This is a clipping mask.
Now, you probably need a compound clipping mask.
A compound path is an object with 2 or more paths, such as the letter 'o'. There's an outer path and an inner path.
In your case, make one box around all of the color, then make another box in the center, or whatever shape you need to knock out. Select both new objects and Object > Compound Path > Make.
Now, select your new compound path and the artwork and create a clipping mask, using the steps in the previous paragraph.
Let us know if you have any issues.
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