When you have multiple downstream dependencies that are accessed via HttpClient in .NET then you want a simple way of logging all the error responses that are received from those calls. Often your application will react to the non-successful response by logging it's own error, but this can sometimes miss the detail of what actually went wrong downstream. An easy to way to capture all the non-successful outbound calls your application makes is to inject a custom delegating handler into all instances of HttpClient (via DI) which can inspect the return code and call out to your logger if necessary: LoggingMessageHandler.cs:
public class LoggingMessageHandler : DelegatingHandler
{
	private readonly IExceptionLogger _exceptionLogger;

	public LoggingMessageHandler(IExceptionLogger exceptionLogger)
	{
		_exceptionLogger = exceptionLogger ?? throw new ArgumentNullException(nameof(exceptionLogger));
	}

	protected override async Task<HttpResponseMessage> SendAsync(
		HttpRequestMessage request, System.Threading.CancellationToken cancellationToken)
	{
		var response = await base.SendAsync(request, cancellationToken);
		if (!response.IsSuccessStatusCode)
		{
			var responseContent = await response.Content.ReadAsStringAsync();
			await _exceptionLogger.LogExceptionAsync(new HttpRequestException(
				$"Api call to {request.RequestUri} failed with status code {(int)response.StatusCode}. Response content: {responseContent}"));
		}
		return response;
	}
}
Your implementation of "IExceptionLogger" may vary, but this is your opportunity to write something to your logs/DB etc. To set this delegating handler on your HttpClient, setup the Microsoft DI container as follows:
services.AddHttpClient<ISomeClient, SomeClient>().AddHttpMessageHandler<LoggingMessageHandler>()
Since this class and custom logic will typically live in your composition root, I'd recommend passing the builder delegate down to any class libraries you are building (where you choose to define the ServiceCollectionExtensions inside the class library). For example: Startup.cs:
services.AddMyCustomLibrary(builder => builder.AddHttpMessageHandler<LoggingMessageHandler>());
CustomLibrary/ServiceCollectionExtensions.cs:
public static void AddMyCustomLibrary(this IServiceCollection services, Action<IHttpClientBuilder> clientBuilderDelegate = null)
{
	var someInnerClientBuilder = services.AddHttpClient<ISomeInnerClient, SomeInnerClient>();
	clientBuilderDelegate?.Invoke(someInnerClientBuilder);
}