Peninei Halachah brings a citation from the Rama 565:3 that we consider that an individual might be forced to break the fast before having fasted most of the day. Once he has fasted until mincha, then he can say aneinu (as long as he is still fasting). Even if he must break the fast after mincha, he has still fasted sufficiently to establish the day as "a day of fasting" and it will not be a lie.
Ashkenazi Jews are accustomed to saying Aneinu in Minchah alone,
because they are concerned that perhaps someone will say it in
Shacharit, become weak during the day, and break his fast. Then, his
statement “on this day of our fast” will turn out to be a lie.
Therefore, they say Aneinu only in Minchah, because one who has fasted
this long will probably complete the fast (based on the Geonim and
Rashi; Rama 565:3).
There are those who say that even if someone has already broken his own fast, he can still say aneinu in the mincha shmona esrai. Others (see Rabbi Kaganoff below) say that only someone still fasting can say it.
Note also that even though your own fast is broken you can still add
the aneinu prayer in the mincha services. Since most people are still
fasting, you can still refer to the day as “the day of our fast”
(Shulchan Aruch 568:1, MB 3).
The chazan can say it during Shacharis because the community as a whole will definitely be able to finish the fast. Since he is the representative of the community, he is able to say it without the fear that applies to an individual.
Note that the chazan says it before Refaeinu and that is the significance of his saying it for the community (and not for himself). As a result, he adds it at the point that the Chazal stated he should in order to precede the request for an answer to the prayer for communal healing. An individual adds it to Shma Koleinu at mincha because it is an individual prayer, just as he would add a prayer for any private purpose. Rabbi Kaganoff points out that the Chazan does not say aneinu during the silent Shmona Esrai of Shacharis. If the Chazan forgets it before Refaeinu, he says it in Shma Koleinu just as an individual does.
It should be said before the blessing “Refa’einu.” If the shaliach
tzibbur forgot but remembered before saying God’s Name in the
conclusion of the blessing, he goes back and says both Aneinu and
Refa’einu. If he has already said God’s Name, he must complete the
blessing of Refa’einu and insert Aneinu in the blessing of Shomei’ah
Tefilla, like an individual does. If he forgot it there, too, he says
it after the Shemoneh Esrei without a blessing conclusion. (Mishnah
Brurah 566:13 says that if there are not ten fasting, the shaliach
tzibbur says Aneinu in Shomei’ah Tefilla, like an individual.)
WHEN DOES ONE RECITE ANEINU?
There is a difference in practice between Ashkenazim and Sefardim.
Sefardim recite Aneinu in all the prayers of a fast day, even the
Maariv of the night before. Ashkenazim recite Aneinu only at Mincha,
except that the Chazon recites Aneinu in the repetition of Shmoneh
Esrei in Shacharis (but not in his private prayer).
Only someone who is fasting recites Aneinu (Maamar Mordechai and Biyur
If someone forgot Aneinu in Shmoneh Esrei, what does he do?
If he is still in the middle of the bracha of Shma Koleinu, he should
recite Aneinu and then complete “Ki atah shomeah” and the bracha. If
he has completed the bracha, he does not repeat any part of the
tefilah. Instead, he recites Aneinu at the end of Shmoneh Esrei as
part of “Elokai Netzor”, preferably before reciting the pasuk “Yihyu
l’ratzon” (Mishneh Berurah 565:6,7; Kaf HaChayim 565:3).