You could try something like this:
string xmlInput = @"<Snippets>
<Snippet name=""abc"">
<SnippetCode>
code goes here
</SnippetCode>
</Snippet>
<Snippet name=""def"">
<SnippetCode>
code goes here
</SnippetCode>
</Snippet>
</Snippets>";
// create the XML, load the contents
XmlDocument doc = new XmlDocument();
doc.LoadXml(xmlInput);
// find a node - here the one with name="abc"
XmlNode node = doc.SelectSingleNode("/Snippets/Snippet[@name="abc"]");
// if found....
if (node != null)
{
// get its parent node
XmlNode parent = node.ParentNode;
// remove the child node
parent.RemoveChild(node);
// verify the new XML structure
string newXML = doc.OuterXml;
// save to file or whatever....
doc.Save(@"C:\temp\new.xml");
}