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In Inkscape the canvas has no fill by default. The canvas is transparent despite being shown as white on-screen. You can see this transparency in Inkscape if you go into File > Document Properties and set the "Chequerboard background" option. Exporting an SVG as PNG will therefore result in areas with no objects being rendered as completely transparent.
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This can be done in Inkscape relatively easily, and the following method is similar to how the Shape Builder tool works in Adobe Illustrator.
Make sure everything is ungrouped, and there is no clipping mask just a regular circle instead to be used to cut everything out.
Make a combined path of the drawing plus the circle, by using Path > Combine Ctrl/...
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Hit Shift+O and adjust your artboard to only include the area you need to export
Do a File → Save for Web and uncheck the Clip to Artboard option.
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Select > Select All
Type > Create Outlines
File > Export
Edit > Undo Create Outlines
By creating outlines you eliminate the path the type is sitting on and the export won't have any problems.
Be certain to Undo after the export... or copy/paste to a new file then create outlines and export and then merely don't save the new file.
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