Two blessings of Shmona Esrei mention grace (חנן) while the rest do not. The fourth blessing (אתה חונן) mentions this root three different times. The sixth bracha (סלח לנו) concludes with it . Why do these brachot rely on G-d's grace or free and undeserved gift, whereas the rest do not?
Regarding the 1st bracha, Chonen Hada'at, see answers to this question. The focus and title of the question is different, but I think the answers, there, address your question. If not, please inform me.
Regarding the 2nd bracha "Selach Lanu", Excerpting some ideas from this article:
Nusach Eretz Yisra'el does not have the word "Chanun" at the ending. This started to appear from the times of Siddur Rav Amram Ga'on. Why was this inserted?
- Abudraham - Aligns with the theme in Tehilim 148:8 "G-d is gracious and merciful"
- Siddur Hagr"a - According to theme in the verse from Shemot 33:19, "I will be gracious to whom I shall be gracious and have mercy on those I shall have mercy." This means that G-d is gracious to a sinner the 1st time he sins even when He knows that he may sin again.
- There was a time when the 13 Middot and Selichot were inserted at this point during fast days (See Shulchan Aruch 566:4). See p. 2 of the article for explanationon why the concept of Chanun became part of Selichot, which would explain, in terms of how it got into the bracha, itself.