你可以试试 DateTime.DayOfWeek 跳过所有已经过的日期 DayOfWeek.Wednesday 或者 DayOfWeek.Thursday 这样地：

for (DateTime date = startdate; date <= enddate; date = date.AddDays(1))
{
    if (date.DayOfWeek == DayOfWeek.Wednesday || date.DayOfWeek == DayOfWeek.Thursday)
    {
        // skip to next day
        continue;
    }

    try 
    {
        // write to database
    }
    catch (Exception)
    {
        Console.Write("not entered");
        //lblError.Text = ex.Message;
    }
}

datetime具有dayofweek属性(看看这里）。
你可以在if语句中使用它，你就可以开始了。

我没有根据星期三和星期四这两个工作日制定条件，而是根据其他5天编写了条件，如下所示。

protected void Submit_click(object sender, EventArgs e)
{
    DateTime startdate = Convert.ToDateTime(txtstartdate.Text);
    DateTime enddate = Convert.ToDateTime(txtenddate.Text);
    for (DateTime date = startdate; date <= enddate; date = date.AddDays(1))
    {
         if (date.DayOfWeek == DayOfWeek.Monday || date.DayOfWeek == DayOfWeek.Tuesday || date.DayOfWeek == DayOfWeek.Friday || date.DayOfWeek == DayOfWeek.Saturday || date.DayOfWeek == DayOfWeek.Sunday)
         {
              var shtdate = date.ToShortDateString();
              string MyConString = "SERVER=localhost;DATABASE=mydb;UID=myid;PASSWORD=abc123;";
              MySqlConnection connection = new MySqlConnection(MyConString);
              string cmdText = "INSERT INTO approved(agentlogin ,leavetype ,date ,time, reason)VALUES ( @login, @type, @date, 'Full day', @reason)";
              MySqlCommand cmd = new MySqlCommand(cmdText, connection);
              cmd.Parameters.AddWithValue("@login", Label1.Text);
              cmd.Parameters.AddWithValue("@type", ddlleavetype.Text);
              cmd.Parameters.AddWithValue("@date", shtdate);
              cmd.Parameters.AddWithValue("@reason", txtreason.Text);
              connection.Open();
              int result = cmd.ExecuteNonQuery();
         }
         else{ 
         }
    }

}