MarkMonitor is a company that helps other companies protect their brands, so they are likely simply scanning your site (not downloading the content for use elsewhere). If you find that there bot is making too many requests of your site, or you simply do not wish to have them scan your site, you can block the IP of their both using the IP Deny Manager in cPanel. I'm not sure what you might be referring to by a "cookie that they left" as cookies are created by the server to be used in browsers and wouldn't be placed on your server by a bot scanning your site.
Please let us know if you have any additional questions or need further assistance.
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