Cut + paste requires different privileges, except copy + paste. You do not specify what you need.
For copy + paste, assign your roles in Contributor + Editor to insert the folder into the sound. I think that in addition to the fact that the user should be able to view the content itself in addition to its folder.
For cut + paste, the user should also be able to delete items from the source folder, which may be the cause of your problem here (cut + paste is implemented as "delete" from the source, and "add" to the destination).
As Auspex says, check the restrictions on the subfolder.
You can run Plone in the "details" mode, which should help find the problem. See http://plone.org/documentation/kb/debug-unauthorized . You will probably need instructions in the "For Zope 2.8 +" section.
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