I think that this is what you are after:
Code:
$sErrorMessage=""
For Each $File in $Files
	If $File
		;Split the time at the : character and then join the parts with no separation characters. 
		$Time = Join(Split(@TIME,':'),"")
		;Copy the file and rename it during the copy process. 
	 	Copy $File 'X:\$StringUsername\' + $Date + '-' + $Time + '.WMA'
		If @ERROR
			$ERROR=@ERROR $SERROR=@SERROR
			LogAction("File " + $File + " copy failed. Reason: ["+$ERROR+"] "+$SERROR)
			$sErrorMessage=$sErrorMessage+"File: '"+$File+"' could not be copied .Reason: ["+$ERROR+"] "+$SERROR+@CRLF
		Else
			LogAction("File " + $File + " was renamed to: " + $Date + '-' + $Time + '.WMA')
			; *** Uncomment the next line to remove the copied file from the thumb drive ***
			;Del $File
			If @ERROR
				LogAction("File " + $File + " could not be deleted from source. Reason: ["+$ERROR+"] "+$SERROR)
				$sErrorMessage=$sErrorMessage+"File: '"+$File+"' could not be deleted from source. Reason: ["+$ERROR+"] "+$SERROR+@CRLF
			Else
				LogAction("File " + $File + " deleted from source successfully.")
			EndIf
		EndIf	     	
	 	;Sleep 1 second to get the $Time var different for each file.
	 	Sleep 1
	EndIf
Next
 
If $sErrorMessage
	$vDiscard=Messagebox("The following files errors occurred during copy:"+@CRLF+@CRFL+$sErrorMessage,"Error during copy")
EndIf


As each file is copied it is deleted if the copy was successful. If the copy fails (or the delete) the process will continue, leaving the failed copy on the source drive.

At the end you will get a pop-up showing failed copies or deletes.

You might want to add some code to delete the copied file if the delete from the source fails, but I would take the safer path and leave it there.

I've commented out the file delete until you are certain that it is all working.

BTW, the code is completely untested, so apologies for any typos.