Remove empty/blanks elements in collection of XML nodes

A single one-liner could do the job, no need to iterate over all elements. Here it goes:

doc.Descendants().Where(e => string.IsNullOrEmpty(e.Value)).Remove();

Tester

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Xml;
using System.Xml.Linq;

namespace ConsoleApplication1
{
    public class TestRemove
    {
        public static void Main() {
            Console.WriteLine("----OLD TREE STARTS---");
            XElement doc = XElement.Parse(@"<magento_api>
                                              <data_item>
                                                <code>400</code>
                                                <message>Attribute weight is not applicable for product type Configurable Product</message>
                                              </data_item>
                                              <data_item>
                                                <code>400</code>
                                                <message>Resource data pre-validation error.</message>
                                              </data_item>
                                              <data_item>
                                                <code>1</code>
                                                <message></message>
                                              </data_item>
                                              <data_item>
                                                <code></code>
                                                <message>No code was given</message>
                                              </data_item>
                                        </magento_api>");
            Console.Write(doc.ToString());
            Console.WriteLine("");
            Console.WriteLine("----OLD TREE ENDS---");
            Console.WriteLine("");
            doc.Descendants().Where(e => string.IsNullOrEmpty(e.Value)).Remove();
            Console.WriteLine("----NEW TREE STARTS---");
            Console.Write(doc.ToString());
            Console.WriteLine("");
            Console.WriteLine("----NEW TREE ENDS---");
            Console.ReadKey();
        }
    }
}

And it also could be tested here