How to manually (via code) apply DbMigration which was manually crafted?

It doesn't make sense to cherry pick a migration and run it, because the migrations are cumulative and must be run in sequence. As such, you'd be better to run the equivalent of update-database powershell command at application startup.

Here's some code we use to do that:

In the Configuration.cs class constructor (this file was made when you enable-migrations)

AutomaticMigrationsEnabled = false;
AutomaticMigrationDataLossAllowed = false;

then at app startup call the following method:

public static void ApplyDatabaseMigrations()
{
    //Configuration is the class created by Enable-Migrations
    DbMigrationsConfiguration dbMgConfig = new Configuration()
    {
        ContextType = typeof(MyDbContext) //+++++CHANGE ME+++++
    };
    using (var databaseContext = new MyDbContext()) //+++++CHANGE ME+++++
    {
        try
        {
            var database = databaseContext.Database;
            var migrationConfiguration = dbMgConfig;
            migrationConfiguration.TargetDatabase =
                new DbConnectionInfo(database.Connection.ConnectionString,
                                     "System.Data.SqlClient");
            var migrator = new DbMigrator(migrationConfiguration);
            migrator.Update();
        }
        catch (AutomaticDataLossException adle)
        {
            dbMgConfig.AutomaticMigrationDataLossAllowed = true;
            var mg = new DbMigrator(dbMgConfig);
            var scriptor = new MigratorScriptingDecorator(mg);
            string script = scriptor.ScriptUpdate(null, null);
            throw new Exception(adle.Message + " : " + script);
        }
    }
}