The dj wanted to play a variety of contents or songs, the radio station's owner did not want even though he claimed was an expert.

C,E,I are the answers

how could (eclectic) is better choice than (homogenous)

The word "mix" signals something that is diverse, so we use the word eclectic, which means assorted. Homogenous means something that is uniform, so it does not make sense within the context of the sentence.

The newly hired DJ petitioned the management to play a more (i)________ mix of music; indeed, the radio station had once been known for playing a wide range of styles. Her manager, while sympathetic, was (ii)________ by the station’s owner, who insisted on only pop music, despite his (iii)________ as a connoisseur of several other genres.

So the Dj is asking the station to play more a electic ( wide range ) of music as radio station once has been known for playing such variety. Her manager who sort of agreed to this recommendation was impeded or obstructed by the station's owner who insisted only on pop music, despite he was renown or famous for his wide variety of taste.