I want to confirm with you, do you mean that you could send email from Exchange on-premises mailbox to Exchange online mailbox successfully?
If so, I would suggest you try to Use Telnet to test SMTP communication on Exchange servers. If you cannot relay email with telnet, it means there exits issue with your Exchange configuration.
In a coexist environment, you need to use Exchange 2013 as Internet facing server, which means you need to point all DNS records(mail.domain.com, autodiscover.domain.com and other records) to Exchange 2013 rather than Exchange 2010. After that, rerun HCW to choose Exchange 2013 as hybrid server. Then try to relay email again.
If you could relay email from telnet successfully, it means this issue may related with the configuration on your Application. Please have a check with it.
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