What you're looking for is ruled in Shulchan Aruch Orach Chaim 591:2. The Shulchan Aruch writes:
המנהג פשוט בכל בני ספרד שאין מזכירין פסוקי קרבן מוסף כלל
The widespread custom in all the Sefardic countries is not to mention any verses of the Mussaf Offering
This is referring to not saying both the Rosh Hashanah Mussaf offerings and the Rosh Chodesh Mussaf offerings.
Whereas, the Rema there rules:
והמנהג הפשוט באשכנז ובגלילות אלו לומר פסוקי מוסף ראש השנה ואין אומרים פסוקי מוסף ראש חודש אלא אומרים מלבד עולת החודש ומנחתה ועולת התמיד ומנחתה וכו' ושני שעירים לכפר ושני תמידים כהלכתם
And the widespread custom in Ashkenaz and the surrounding areas is to say the verses of Mussaf for Rosh Hashanah, but not to say the verses of Mussar for Rosh Chodesh. Except for "the Olah offering of the New Month, and its Mincha offering, and the daily Olah offering, and its Mincha offering...and two goats to atone and two daily offerings according to their laws".
Now, what's the reason for this? (Their dispute is the topic for a different question) The Rema's custom comes from Rabbeinu Tam, brought by Tosafos to Rosh Hashanah 35a s.v. אילמא who explains:
מקראות של מוסף צריך להזכיר לעולם שהם במקום הקרבנות אבל מוסף דר"ח אין צריך להזכיר בר"ה אלא שדי לו במה שאמר מלבד עולת החדש ומנחתה שבזה נכלל כל מוספי ר"ח העולות וכדי להזכיר גם השעיר שהוא חטאת יאמר ושני שעירים לכפר ושני תמידים כהלכתן
The verses of Mussaf always need to be mentioned, as they're in place of bringing the offerings. However, the Mussaf of Rosh Chodesh doesn't need to be mentioned on Rosh Hashanah. Rather, it's sufficient to say "besides the Olah of Rosh Chodesh and its Mincha offering", as this includes all the Mussaf offerings of Rosh Chodesh that are an Olah. As well, in order to mention the goats which are a Chatas, say "and the two goats for atonement, and the two daily offerings according ot their laws".
Now, he doesn't fully explain why not mention the verses. He just says that the verses of Rosh Hashanah allude to the offerings of Rosh Chodesh, and it's sufficient to say those verses (presumably for brevity).
The Bach ad. loc. tries to provide a more detailed explanation:
נראה לומר דמה שאינו מזכיר של ר"ח בפירוש ה"ט מדאסיקנא זכרון אחד עולה לכאן ולכאן פי' שאינו אומר את יום ר"ח הזה כו' אלא את יום הזכרון הזה לבד ומשמע דלכל ענייניו קאמר דזכרון דר"ה עולה לר"ח לפיכך בהזכירו מקראות דמוסף דראש השנה ואומר בלבד ג"כ חשיב ליה כאילו הכל נכלל בזכרון אחד...
It appears that the reason why don't explicitly mention [the Mussaf offerings] of Rosh Chodesh is for the following reason: Since we rule that the mentioning of one of the holidays suffices for both. Meaning we don't mention "this day of Rosh Chodesh", rather, we say "this Yom HaZikaron" alone, which implies all of its aspects. That is, the mentioning of Rosh Hashanah suffices for Rosh Chodesh. Therefore, when we mention the verses of Mussaf for Rosh Hashanah, and we say "besides...", this also is considered as if everything is included in this reference...
The Beur HaGra ad. loc. also cites Tosafos to Rosh Hashanah 8b who s.v. שהחודש מתכסא write:
ועוד פי' מתכסה שאין מזכירין מוסף של ר"ח בתפלה
Another explanation for "covered", is that we don't mention the Mussaf of Rosh Chodesh in our prayers.
This is referring to what the gemarra there says, to explain the verse in Psalms 81:4,5:
תקעו בחדש שופר בכסה ליום חגנו כי חק לישראל הוא משפט לאלהי יעקב
Blow a shofar at the New Month, the one that is covered, for the day of our Holiday. It is a decree for Israel, it is a judgement for the G-d of Yaakov.
The gemarra asks what does it mean that it is covered? It refers to the holiday that is covered, namely Rosh Hoshanah. Tosafos explain that it covers the holiday of Rosh Chodesh, meaning that any mention of Rosh Chodesh is minimal at best. If you look at the rest of Tosafos, they sort of bring a reason behind this. It is because דמוטב שיבא זכאי ויכפר על החייב, it is better for someone innocent to come and atone for the one who is guilty (instead of the other way around). Rosh Chodesh is called "guilty", because Hashem (so to speak) needs atonement on Rosh Chodesh for shrinking the Moon, whereas Rosh Hoshanah is called "innocent".
It comes out then that the first two reasons hold it is unnecessary to mention Rosh Chodesh, and the third reason holds its meant to be omitted.
