جَاءَ زَيْدٌ وَهُوَ مَرِيضٌ.
Zayd came while ill.
The illness describes the state accompanying the arrival, supporting a circumstantial reading. In «جَاءَ زَيْدٌ — حَفِظَهُ اللَّهُ — إِلَى الْمَجْلِسِ» ('Zayd—may God protect him—came to the gathering'), the prayer does not describe how he came; it interrupts the report with an independent benefit.
Both added clauses might be removable without destroying grammaticality. The critical test is whether the clause reports a concurrent state or provides a parenthetical comment.