Why We Add ולכפרת פשע on ר"ח of A Leap Year

On Rosh Chodesh  of a leap year we added in the Shemona Esreh of Mussaf  the words  ולכפרת פשע
In a Leap Year, in addition to praying for good and for bracha, for ששון and for שמחה, for salvation and comfort, for פרנסה and כלכלה, for life and peace, for pardon of sin and forgiveness of iniquity”, we add a 13th request in respect of the additional month: “ולכפרת פשע-and for atonement of transgression”.

A possible reason suggested is that in case the Leap Year was mistakenly designated, and in consequence Chametz was eaten on Pesach, we pray for forgiveness. (טעמי המנהגים)
The 13 pleas are corresponding to the 12 expressions of praise “הללוי’ה-הללו-הללוהו”, and the last sentence “כל הנשמה” is said twice, in respect of the Leap Year. (תהלים קנ – תניא רבתי)

- (Mi Yodea) There are three different minhagim (according to Kaf HaChayim) for the timing of this addition: only during Adar 2, all year long (Cheshvan to Elul), and from Cheshvan to Adar 2.

Rabbi Shelomo Min-HaHar z"l (former rabbi of the Bayit VaGan community in Jerusalem) explains that this last minhag (which you're asking about) is due to the fact that from Nissan onward one cannot declare that year as a leap year, so that's when the addition of "ולכפרת פשע" stops.

Given that there are 3 possibilities as brought down in the Mishna Brurah 423 or the Kaf Hachaim. However, the prevalent minhag among Ashkenazim is from חשון to אדר שני.

הוספת "ולכפרת פשע" בשנה מעוברת

אבל המנהג הרווח הוא, לפי הספרדים כל השנה, ולפי האשכנזים מתחילת השנה עד חודש העבור.

The prevalent minhag among the Sefardim is the entire year and among the Ashkenazim until the leap month (Adar II)

This explains the Ashkenaz minhag rationale.

Since Adar II is the last month of the year when counting by months, and Rosh Chodesh Nisan is the first month then the minhagim of Rosh Chodesh would follow that way of counting. Thus, the addition of ולכפרת פשע would end at the "end of the year" by months.

Even if you count by the change in the year number (Tishrei), by the end of Adar II, the leap month has been added and the months have been reset to synchronize with the seasons. If the Sanhedrin were to determine the leap year, the end of Adar would be the absolute last time it could have been done. As a result, we have now added the thirteenth month and the brachos for the months now revert to the normal 12. Once Rosh Chodesh Nisan has passed (without an extra month having been set to synchronize it) a leap year cannot be decreed until the following Adar.

It should be noted that the gemara mentions that Chizkiyahu decreed a second Adar at Rosh Chodesh Nisan and the Chachamim stated that he was wrong to do so. See Sanhedrin 12b.

The question would more likely be "why start saying it in Tishrei rather than the previous Nisan" if you count by Nisan as the first month.

ראש חודש הוא זמן כפרה על מיעוט הירח, כמו שדרשו בגמרא על שעיר ראש חודש 'הביאו עלי כפרה על שמיעטתי את הירח',

כל עיבור השנה נגרם ממיעוט הירח, שהרי אם היה הירח גדול כחמה לא היה הפרש ביניהם. כל ההפרש של י"א ימים בין שנת חמה ושנת לבנה, הוא מכח שנתמעט הירח.כאשר מעברים את השנה, מתקנים את מיעוט הירח במידה מסויימת, ועל פי הסוד זה התיקון של גלות השכינה הנרמזת בלבנה. מכיון שהקב"ה אמר 'הביאו עלי כפרה', מוסיפים 'ולכפרת פשע' כביכול כלפי הקב"ה, ואומרים  זאת בשנת העיבור שבו הוא בעיקר זמן התיקוןיותר   מכל השנים