Here is the process when you send emails through Kennel Connection.
When you hit Send or it's time to send automatic emails, KC logs into your email provider's server. If your login information is authenticated, KC sends all emails to the server. That's all it does. It is up to your provider to send the emails to your clients.
This error means that KC has sent the email successfully to your email provider. When your provider attempts to send this to your client, the message is being rejected by your client's server, saying the email doesn't exist. Verify with your client that you have the correct email address listed. Be sure there are no spaces before or after the address. If that does not fix it, you will need to work with your client and their provider to determine why it won't accept emails from you.