Some users belong to a group where the Move and Delete button are missing on GUI. This groups where copied from other groups who have that options so not sure where this can be added/edited ?
Do you know where this options can be added ( move and delete function) to a particular group only ?
Can you please verify that the user group have permissions to "Delete" a certain resource within AEM. A common cause of the missing "Delete" and "Move" button will be because of the usergroup/user does not have ACL permissions to delete.