谢谢@ Alexandria ，
在你的评论之后，我再次检查了我的代码，发现这个问题不是由await base.SendAsync（）引起的。事实上，我是用异步调用内部方法，这破坏了HttpContext.Current。
下面是我的DelegatingHandler实现，作为参考。在此之前，我定义了两个方法，即IncomingMessageAsync()和OutgoingMessageAsync作为async。现在我已经将其更改为简单方法。因为我不希望在这些方法中使用Task。

public  class MessageHandler : DelegatingHandler
{
    private static NLog.Logger logger;
    HttpContext context;

    public MessageHandler()
    {
        logger = NLog.LogManager.GetCurrentClassLogger();
    }

    protected override async Task<HttpResponseMessage> SendAsync(HttpRequestMessage request, CancellationToken cancellationToken)
    {
        var correlationID = $"{DateTime.Now.Ticks}{Thread.CurrentThread.ManagedThreadId}";
        context = HttpContext.Current;
        context.Items["correlationID"] = correlationID;     //used in Nlog aspnet-item:variable=correlationID
        IncomingMessageAsync(request);
        var response = await base.SendAsync(request, cancellationToken);
        context.Items["responseTime"] = DateTime.Now;
        HttpContext.Current = context;
        OutgoingMessageAsync(response);
        return response;
    }

    protected void IncomingMessageAsync(HttpRequestMessage request)
    {
        logger.Info("Request received from {URL}", request.Method + " " + request.RequestUri);
    }

    protected void OutgoingMessageAsync(HttpResponseMessage response)
    {
        if (response.IsSuccessStatusCode)
        {
            logger.Info("Response returned with status {Status}", response.StatusCode);
        }
        else
        {
            logger.Warn("Response returned with status {Status}", response.StatusCode);
        }
    }
}