The underscore character, when used in the search dialog, acts in what appears to be an undocumented fashion (I couldn't find anything, and nor could Esri UK support), as illustrated in the screenshot below. Support suggested that I add an idea here to suggest that the documentation be updated to cover what the underscore character 'means' when used in the search dialog. Replacing the underscore with a question mark enabled me to get the results set I wanted.
Search term: *09_geom*. Search returns items with 09 and geom as separate components of their name, rather than items with 09_geom as a single component of their name.