As a partner, if you added your customer's domain to Cloudflare via the Host API, you are able to manage their domain settings using the Client API.
Please note that the user account has to have been created through the Host API or one of the associated plugins (e.g. the Cloudflare plugins cPanel or Plesk). The Host API will return the user_key and user_api_key when employing the 'user create' call.
'user_api_key' is used exclusively with the Client API to manage domain settings.
If your customer has updated the password after you created the account, you are still able to manage domain settings with the 'user_api_key'.