Often clip art is drawn as a series of stacked stroked and filled outline, e.g.,
which looks line, but is a confusing mess when one views it in outline mode:
The solution is to expand the strokes, and punch out the white fills. In Inkscape this is done by:
Path | Object to Path
Then break everything apart (it is important to work in View | Display Mode | Outline) using Path | Break Apart
Then select all of the outer black outlines (you may need to ungroup things to ensure that one is only selecting paths) and union them:
Path | Union
Then select each of the paths which define the white regions and either punch them out of the other path or union them as needed to create the desired appearance.
Depending on the complexity it may be helpful to duplicate the original paths to a layer and use them to punch things out at need.