11. Because the dog had problems that only its analyst knew and were quite embarrassing and because the dog had been a friend to the emperor, the kennel master sought additional counseling for the Pekingese that was fixated on sniffing the particulars of the dachshunds, terriers, and the other caged animals.

Ans.

11. Because the dog had problems that only its analyst knew and that were quite embarrassing and because the dog had been a friend to the emperor, the kennel master sought additional counseling for the Pekingese that was fixated on sniffing the particulars of the dachshunds, terriers, and the other caged animals.

Despite all of the parallel constructions here, the only one that needs to be fixed is the compound relative clause near the beginning of the sentence. The relative clause "that only its analyst knew" and "that were quite embarrassing" are joined together as a compound relative clause. The second of these two needs the relative pronoun "that" to distinguish the compound construction.

Notice that the relative clause "that only its analyst knew and were quite embarrassing" cannot employ a compound verb (as in "its analyst knew and were . . .") because the nature of these two verbs is different. "Knew" is an "action" verb and "were" is a form of to be.