A customer cannot remove an Exchange public folder with child public folders in the POA Customer Control Panel - there is no way to select such public folder for removal, e.g. on the attached screenshot it may be seen that the 'pf1' and 'pf2' folders cannot be selected:
This behavior is by design. A customer has to remove all child public folders first in order to remove parent public folder. On the screenshot above after customer remove the 'pf3' public folder a checkbox allowing to remove the 'pf2' public folder will appear. When the 'pf2' is deleted it will be possible to delete the 'pf1'.
If some child public folder cannot be deleted (stuck in the 'Deleting' status) you will have to locate a task to remove it in POA Provider CP under System Director > Task Manager > Background tasks and if the task failed fix the reason why it failed and process the task. Task to remove Exchange public folder is named as 'Remove public folder (id=)' where is the name of public folder to remove and is its ID which may be seen in POA Provider and Customer Control Panel.
You may filter tasks related to a particular subscription using the 'Subscription ID' field in the POA Task Manager. All Exchange related tasks for a subscription will be put in the same task queue, so it is possible that a task 'Remove public folder' can be blocked by another failed or non-processed Exchange task(s) for the same subscription. In this case it will be necessary to process that failed task(s) first before task to remove the public folder can be executed.