You can prepend string.Empty to an existing list with concat:
var emails = new List<string> {"[email protected]", "[email protected]"};
policy.Emails = new[] {string.Empty}.Concat(emails).ToList();
Now policy.Emails looks like this:
{"", "[email protected]", "[email protected]"}
If you would like to replace the first item, use Skip(1) before concatenating:
policy.Emails = new[] {string.Empty}.Concat(emails.Skip(1)).ToList();
To generalize, replacing the initial n values with empty strings would look like this:
policy.Emails = Enumerable.Repeat(string.Empty, 1).Concat(emails.Skip(n)).ToList();
Note: It goes without saying that if you do not mind modifying the list in place, the simplest solution is to do
emails[0] = string.Empty;