Learn Turkish: Lütfen yeni genelgeyi ve duyuruyu oku. - Please read the new circular and the announcement.

Because they are both specific direct objects of the verb oku (read). When you have multiple definite objects connected by ve (and), each noun gets its own accusative case marker.

Why is there a y before the ending in both genelgeyi and duyuruyu?

Both root words end in a vowel (genelge and duyuru). The accusative case suffixes are also vowels (-i and -u). Turkish uses the buffer letter y to prevent two vowels from crashing into each other.

Does the adjective yeni (new) apply to the announcement as well as the circular?

Just like in English, it is a bit ambiguous. Grammatically, it sits right before genelgeyi, so it definitely applies to the circular. If you wanted to be completely clear that both are new, you would say yeni genelgeyi ve yeni duyuruyu.

How is the word duyuruyu formed?

The root is the verb duymak (to hear). From that comes duyurmak (to make heard / to announce), which turns into the noun duyuru (announcement). Finally, we add the buffer y and the accusative suffix -u to get duyuruyu.