Multiple MX Records handling
MX records are used according to priority value in the records.
The record with the lowest priority is used first, then the higher ones until one responds. If there are multiple records with the same priority, one is randomly selected (this is how you generally do load balancing if you have multiple mail servers accepting incoming connections).
The MX records only dictates which mail servers are responsible for a specific domain, it doesn't deal with individual recipients. So a sending server will only use secondary records if the primary server doesn't respond to its connection attempts, not if the primary server rejects the message.
What you're trying to achieve is only doable at the DNS level if you use a subdomain for messages destined for your application. That way you can have the MX records for
example.com point to your mail servers and the MX records for
app.example.com to point towards your application.
If you need to use the same domain for both, you'll need to configure your mail server to forward e-mail messages to your application. This can usually be done a couple of different ways depending on the mail server/hosting provider.
MX records cannot solve this, multiple records (with possibly different priorities) can be used for redundancy, but the service is expected to be the same (accept the same addresses).
What you can do is either have the address that your application processes at a different domain (eg
[email protected] if the regular addresses are
@example.com) or set up some forwarding solution with the other mail service (eg forward
[email protected] to wherever you can deliver mail for the application).