Problem with loading images from child theme CSS file with relevant path

There’s no need for the wp-content/themes path – both themes sit in the same directory, so you can just traverse up one and then back down to child theme:

background-image: url(../child-theme-name/images/some-image.png);

Update: Regarding your answer to “where is the CSS file stored”, you inferred that the stylesheet resides in the child theme folder – in which case you are massively overcomplicating things and can just use (as @Rishabh suggested):

background-image: url(images/some-image.png);

Relative paths in a stylesheet are relative to the stylesheet itself – not the document, the parent theme, or anything else for that matter.