We will face “INSUFFICIENT_ACCESS_OR_READONLY, insufficient access rights on object id” because of Sharing Settings.

To overcome this issue, check the below

1. Organization Wide Default(OWD), Sharing Rules and Role Hierarchy of the object in which you are trying to update the record.

2. Owner of the record which you are trying to update.

3. Check the trigger logic and check Organization Wide Default(OWD), Sharing Rules and Role Hierarchy of the related objects records which are getting updated.

Note:

Even though trigger runs in System Mode, Sharing Settings will be checked. Only CRUD and FLS will not be checked against the user.