I’m pretty sure that you know that the first Avinu Malkenu was recited by Rabbi Akiva at a fast day for rain. They prayed and were not answered. After the service, Rabbi Akiva descended to the Biman and said Avinu Malkenu, and immediately the sky darkened and it started raining.
In Avinu Malkenu we declare that we speak to Hashem who is both our Father, loving and caring, and Malkenu, our King whose power is endless and who can act based solely on His desires. We ask Him to combine these attributes of love and majesty and to deal with us using His infinite mercy.