You can more easily create the "starburst" effect with a stroked path by drawing one path and using
Effect > Distort & Transform > Transform:
Angle field, merely enter
X is the number of copies you want. Illustrator will do the math for you and determine what angle is necessary to fit
X number of copies into a circle (360°). In the image above, I entered
360/50 and AI determined that 7.2° was necessary.
If you merely want to hide the lines, offset from the primary figure, you can add a white stroke (or whatever color matches the background) to the figure. The fact that things exist under each other shouldn't be an issue.
If you do indeed want to remove the paths with an offset, you could use a Clipping Mask.
Draw a path which encompasses all the "starburst" outside points. Then draw a path which is offset from the primary figure as far as you'd like this "gap" to be. Select both these paths and choose
Object > Compound Path > Make from the menu.
Now select this compound path and the starburst lines and choose
Object > Clipping Mask > Make from the menu.
This will simply hide the portions of the "starburst" which are not within the containing path.
If you really want to just remove the strokes around the primary figure, you are going to have to do some expanding to a degree.
You can take the clipping mask version above -- select the clipping mask and choose
Object > Expand from the menu. Then merely hit the
Crop button on the Pathfinder Panel. However, this will expand the strokes of the "starburst".
You can take a step back... Select the starburst (as created with the effect above) and choose
Object > Expand Appearance. This will create individual paths for each "burst" and remove the live effect aspect.
Now, using a path that is offset from the primary figure, select that and these individual "starburst" paths. Grab the Shape Builder Tool, Hold down the Option/Alt key and click-drag over anything you don't want.
This will remove parts of the strokes offset from the primary figure, but will also keep them as stroked paths.
By keeping the "bursts" as stroked paths, you can easily use the Stroke Panel to assign a profile to the strokes for more variations....
For my work, I generally prefer to use a clipping mask (#3) if a white outline (#2) doesn't suffice. I try not to be destructive and expand things unless I must.