What happens when the same application from different server tries to connect to the always on primary server
From a technical standpoint, there are no problems with multiple applications, running on multiple different machines, connecting to the same SQL Server instance.

If you experience the failover scenario that you described, it's possible that the reporting workload will interfere with your other workloads from a performance standpoint.  

In other words, all the long read queries your reporting app does could block queries coming from your other apps.  And in general the reporting app queries will require CPU time and I/O time that is normally solely used by the other app.

It's hard to say without more detail whether this would just slow things down a little, or bring your whole app grinding to a halt due to blocking or resource contention.