How exactly have you created these "custom integrations"?
Users with Salesforce Platform licences do not and cannot have access to Cases. Those users would need the Salesforce license and those would have to be purchased directly from Salesforce at a higher cost per license.
That said, we did publish an unmanaged package that ties Cases to Incidents. Because we do this through Flows and Process Builder, the staff working the Incident can update the Incident (for example, closing the Incident and updating the Resolution field will in turn update a custom field on the Case object). However, the Staff working the Incident, cannot view or access the original Case.
Thanks for your reply. I am using triggers to update cases when incidents are updtaed by the Service Desk team. However when a person with Salesforce Platform licence tries to update incident, the queries being used in the trigger to fetch case details don't return any rows and integration fails. As soon as I change the licence for Service Desk staff back to Salesforce the integration starts working fine again.
Creating a sharing Rule for the Service Desk Staff team and moving the trigger code to a public with sharing class resolved the issue for me.