To save you the hours of pain and misery, here is some C# code which you can use to call the AWS Polly service, without having to use AWS's bloat-central SDK. You are welcome :)

AWSPolly.cs

using System;
using System.IO;
using System.Net.Http;
using System.Security.Cryptography;
using System.Text;
using System.Threading.Tasks;
using Newtonsoft.Json;

public class PollyService
{
    private const string AccessKey = "<key>";
    private const string SecretKey = "<key>";
    private const string Method = "POST";
    private const string Service = "polly";
    private const string Api = "/v1/speech";
    private const string ContentType = "application/json";

    public static async Task<Stream> TextToSpeech(string textToSpeech, string voice="Brian", string region = "eu-west-1")
    {
        var host = Service + "." + region + ".amazonaws.com";
        var baseUrl = "https://" + host;

        var requestParameters = new RequestParameters
        {
            OutputFormat = "pcm",
            SampleRate = "16000",
            Text = textToSpeech,
            TextType = "text",
            VoiceId = voice
        };

        var t = DateTime.UtcNow;
        var amzDate = t.ToString("yyyyMMddTHHmmssZ");
        var dateStamp = t.ToString("yyyyMMdd");

        var canonicalQueryString = "";
        var canonicalHeaders = $"content-type:{ContentType}\nhost:{host}\nx-amz-date:{amzDate}\n";
        var signedHeaders = "content-type;host;x-amz-date";
        var payload = JsonConvert.SerializeObject(requestParameters);
        var requestParametersBytes = Encoding.UTF8.GetBytes(payload);

        using var digest = SHA256.Create();
        var payloadHash = HashToHexString(digest.ComputeHash(requestParametersBytes));
        var canonicalRequest = 
            $"{Method}\n{Api}\n{canonicalQueryString}\n{canonicalHeaders}\n{signedHeaders}\n{payloadHash}";

        var algorithm = "AWS4-HMAC-SHA256";
        var credentialScope = $"{dateStamp}/{region}/{Service}/aws4_request";
        var canonicalRequestBytes = Encoding.UTF8.GetBytes(canonicalRequest);
        var stringToSign = 
            $"{algorithm}\n{amzDate}\n{credentialScope}\n{HashToHexString(digest.ComputeHash(canonicalRequestBytes))}";
        var key = GetSignatureKey(SecretKey, dateStamp, region, Service);
        var signature = HashToHexString(HmacSHA256(stringToSign, key));
        var authorizationHeader = 
            $"{algorithm} Credential={AccessKey}/{credentialScope},SignedHeaders={signedHeaders},Signature={signature}";

        var client = new HttpClient();
        var content = new StringContent(payload, Encoding.UTF8, "application/json");

        client.BaseAddress = new Uri(baseUrl, UriKind.Absolute);
        var request = new HttpRequestMessage(HttpMethod.Post, new Uri(Api, UriKind.Relative));
        request.Content = content;
        request.Headers.Host = host;
        client.DefaultRequestHeaders.TryAddWithoutValidation("Authorization", authorizationHeader);
        client.DefaultRequestHeaders.TryAddWithoutValidation("Content-Type", ContentType);
        client.DefaultRequestHeaders.TryAddWithoutValidation("X-Amz-Date", amzDate);
        content.Headers.ContentType.CharSet = string.Empty;

        var response = await client.SendAsync(request);
        return await response.Content.ReadAsStreamAsync();
    }

    private static string HashToHexString(byte[] bytes)
    {
        var hex = new StringBuilder(bytes.Length * 2);
        foreach (var b in bytes)
        {
            hex.AppendFormat("{0:x2}", b);
        }

        return hex.ToString();
    }

    private static byte[] GetSignatureKey(string secretKey, string dateStamp,
        string regionName, string serviceName)
    {
        var kSecret = Encoding.UTF8.GetBytes(("AWS4" + secretKey).ToCharArray());
        var kDate = HmacSHA256(dateStamp, kSecret);
        var kRegion = HmacSHA256(regionName, kDate);
        var kService = HmacSHA256(serviceName, kRegion);
        var kSigning = HmacSHA256("aws4_request", kService);
        return kSigning;
    }

    private static byte[] HmacSHA256(string data, byte[] key)
    {
        const string algorithm = "HmacSHA256";
        var kha = KeyedHashAlgorithm.Create(algorithm);
        kha.Key = key;

        return kha.ComputeHash(Encoding.UTF8.GetBytes(data));
    }
}

public class RequestParameters
{
    public string OutputFormat { get; set; }
    public string Text { get; set; }
    public string TextType { get; set; }
    public string VoiceId { get; set; }
    public string SampleRate { get; set; }
}