How to use HttpClientHandler with HttpClientFactory in .NET Core

More properly to define primary HttpMessageHandler via ConfigurePrimaryHttpMessageHandler() method of HttpClientBuilder. See example below to configure typed client how.

services.AddHttpClient<TypedClient>()
    .ConfigureHttpClient((sp, httpClient) =>
    {
        var options = sp.GetRequiredService<IOptions<SomeOptions>>().Value;
        httpClient.BaseAddress = options.Url;
        httpClient.Timeout = options.RequestTimeout;
    })
    .SetHandlerLifetime(TimeSpan.FromMinutes(5))
    .ConfigurePrimaryHttpMessageHandler(x => new HttpClientHandler() 
    {
        AutomaticDecompression = DecompressionMethods.GZip | DecompressionMethods.Deflate;
    })
    .AddHttpMessageHandler(sp => sp.GetService<SomeCustomHandler>().CreateAuthHandler())
    .AddPolicyHandlerFromRegistry(PollyPolicyName.HttpRetry)
    .AddPolicyHandlerFromRegistry(PollyPolicyName.HttpCircuitBreaker);

Also you can define error handling policy via usage of special builders methods of Polly library. In this example policy should be predefined and stored into policy registry service.

public static IServiceCollection AddPollyPolicies(
    this IServiceCollection services, 
    Action<PollyPoliciesOptions> setupAction = null)
{
    var policyOptions = new PollyPoliciesOptions();
    setupAction?.Invoke(policyOptions);

    var policyRegistry = services.AddPolicyRegistry();

    policyRegistry.Add(
        PollyPolicyName.HttpRetry,
        HttpPolicyExtensions
            .HandleTransientHttpError()
            .WaitAndRetryAsync(
                policyOptions.HttpRetry.Count,
                retryAttempt => TimeSpan.FromSeconds(Math.Pow(policyOptions.HttpRetry.BackoffPower, retryAttempt))));

    policyRegistry.Add(
        PollyPolicyName.HttpCircuitBreaker,
        HttpPolicyExtensions
            .HandleTransientHttpError()
            .CircuitBreakerAsync(
                handledEventsAllowedBeforeBreaking: policyOptions.HttpCircuitBreaker.ExceptionsAllowedBeforeBreaking,
                    durationOfBreak: policyOptions.HttpCircuitBreaker.DurationOfBreak));

    return services;
}

Actually I'm not using automatic decompression but the way to achieve this is to properly register http client

services.AddHttpClient<MyCustomHttpClient>()
   .ConfigureHttpMessageHandlerBuilder((c) =>
     new HttpClientHandler()
     {
        AutomaticDecompression = System.Net.DecompressionMethods.GZip
     }
   )
   .AddHttpMessageHandler((s) => s.GetService<MyCustomDelegatingHandler>())