Add HttpMessageHandler to all HttpClients

Startup.cs

services.AddSingleton<IConfigureOptions<HttpClientFactoryOptions>>(provider =>
{
    return new ConfigureNamedOptions<HttpClientFactoryOptions>(name: null, options =>
    {
        options.HttpMessageHandlerBuilderActions.Add(builder =>
        {
            builder.AdditionalHandlers.Add(provider.GetRequiredService<YourHandler>());
        });
    });
});