have you tried this - http://technet.microsoft.com/en-us/library/ee332344(v=exchg.150).aspx

Also you might also use this to move content to different destination
http://blogs.msdn.com/b/emeamsgdev/archive/2012/04/27/powershell-script-to-move-items-from-one-folder-to-another-in-a-user-s-mailbox.aspx

Hello,
I already reviewed the first site.  
The purpose of this script is not to move but to delete a mail item from ALL public folders identified by similiar Subject or From as my example below shows:
Get-PublicFolder | Search-Mailbox -SearchQuery "From:'xxxx'" -LogOnly -TargetMailbox  administrator -TargetFolder "\Public Folders\All Public Folders\"

I can successfully run a smiliar script on the Exchange Mailboxes and it works perfectly.
Get-Mailbox | Search-Mailbox -SearchQuery "From:'XXXX'" -LogOnly -TargetMailbox administrator  -TargetFolder Inbox

When I run a Public Folder item,  I keep getting errors.  

I am trying to do the same thing but to the Public Folders.

Cannot process argument transformation on parameter 'Identity'. Cannot convert the "\" value of type "Microsoft.Exchange.Data.Mapi.PublicFolder Id" to type "Microsoft.Exchange.Configuration.Tasks.MailboxOrMailUserIdParameter".
     + CategoryInfo          : InvalidData: (Microsoft.Excha...pi.PublicFolder:PSObject) [Search-Mailbox], ParameterBin    din...mationException     + FullyQualifiedErrorId : ParameterArgumentTransformationError,Search-Mailbox

I have discovered that deleting items from All Public Folders is not possible unless it is with a VB Script.   With the many Public Folders existing in our organizaiton, I will open a Pay Ticket with Microsoft to see what tricks they have up their sleeves.   I will gladly post any suggestions and resolutions from Microsoft here.     Thank you .