How can I change this image so that the inner circles appear like they are at the highest level? Meaning that the outer and bigger circle doesn't cast a shadow over them?
Edit1: I am using Inkscape
Unclear in your question what app you're using, so I screencapped both Adobe Illustrator and Affinity Designer to show this applies to both for certain, and I'm pretty sure Inkscape and Xara and Sketch as well; It's all down to the draw order - which in Adobe Illustrator and Affinity Designer shows up clearly in the Layers palette.
If you select your smaller circles, and
right click on them, you should get a context menu which says things like "Arrange> Bring/Move Forward", "Arrange> Bring/Move to Front", "Arrange> Bring/Move Backwards" and "Arrange> Bring/Move to Back" - these commands are how you control draw order on an individual graphic element (or multiple selected elements) outside of using the layers palette - you can also grab things in the Layers palette and drag them into a more correct order.
If your little circles are higher in the draw order - hence "above" your other elements, their drop shadows will be cast onto the lower elements as you desire.
Hope this helps.