Xpath to search for a node that has ANY attribute containing a specific string?

The general pattern you're looking for is:

@*[contains(., 'string')]

which will match any attribute on the context element that contains string. So if you simply want to search the whole document for attributes containing string, you'd use:

//@*[contains(., 'string')]

Sample XML:

<root>
  <element1 a="hello" b="world"/>
  <element2 c="world" d="hello"/>
  <element3 e="world" f="world"/>
</root>

Suppose, we need to select elements which have any attribute containing h. In this sample: element1, element2. We can use this XPath:

//*[@*[starts-with(., 'h')]]

In your sample:

/xs:schema/node()/descendant::node()
    [@*[starts-with(@my-specific-attribute-name-here, 'my-search-string')]]