Emails failing in applications after migrating to the cloud

You haven't provided enough information about your setup to advice anything. We have used a relay this way https://docs.microsoft.com/en-us/Exchange/mail-flow-best-practices/how-to-configure-iis-for-relay-with-office-365?redirectSourcePath=%252fen-us%252farticle%252fHow-to-configure-IIS-for-relay-with-Office-365-eb57abd2-3859-4e79-b721-2ed1f0f579c9 so it is an internal server, all applications send emails to it and it securely relays them through EO. But it looks like there are already a few options to do a some what direct sending now (a few years later) https://docs.microsoft.com/en-us/Exchange/mail-flow-best-practices/how-to-set-up-a-multifunction-device-or-application-to-send-email-using-office-3