When can we pray while we're praying?

This was also stated by an amora; Rav Ḥiyya bar Ashi said that Rav said: Although the Sages said that one requests their personal needs in the blessing: Sh'ma Koleinu (Who listens to prayer), if one wishes to do so as part of the Amida prayer. If they come to add and recite additional requests after completing their Amida prayer, even if their personal requests are the equivalent of the order of the confession of Yom Kippur, one may recite them.

With regard to the halakhic ruling, Rav Yehuda says that Shmuel says: The halakha is that a person requests their own needs during the Amida prayer in the Sh'ma Koleinu blessing (Who listens to prayer). Rav Yehuda, son of Rav Shmuel bar Sheilat, says in the name of Rav: Although the Sages said that a person requests their own needs in the Sh'ma Koleinu blessing (Who listens to prayer), that is not the only option. Rather, if they wish to recite at the conclusion of each and every blessing personal requests that reflect the nature of each and every blessing, one may recite them.

(1) The Laws of the One Who Wants To Add [Personal Prayers] In The Blessings. Containing 4 S'ifim:
If one wanted to add in any of the middle blessings, something similar the blessing, one may add. How so? If one had a sick person, one asks for mercy for [that person] in the blessing of "Refa'einu" ["Heal us"].

If one needs a livelihood, one may ask for it in the "Blessing of the Years".

And in [the blessing] of "Shomeya Tefilla" ["Who hears prayers"], one may ask for any of one's needs, for it includes all the requests.

Gloss (R"MA): And when one adds, one should begin the blessing and, after that, add, but one should not add and then begin the blessing (Tur 567).

And according to Rabbeinu Yona, when one adds to the blessing something similar to that blessing, if one is adding it on behalf of all of Israel, one says it in plural language and not singular language, and one should only add at the end of the blessing and not the middle.

And if one is asking specifically for one's own needs, for example: there is a sick person in one's home or one needs a livelihood, one can ask even in the middle of the blessing, as long as one does so in singular language and not plural language.

And in the blessing of "Shomeya Tefilla" and similarly at the end of prayer, either right before "Yihyu l'ratzon" ["May it be acceptable before You"] or after it, one may ask in either singular language or plural language, whether it is specifically for one own needs or for of the public.

(2) There is one [authority] who says that when one adds to a blessing for one's individual needs, one should not make it lengthy.

May it be Your will, ha-Shem, our God, and the God of our ancestors, that You send, quickly, a complete recovery from the heavens, healing for the soul, and healing for the body, for the ailing (name) son/daughter of (mother’s name) among the ailing of Israel.

You are ha-Shem, God who provides sustenance and provides (for all creatures) from the unicorns to the eggs of lice.

Provide me with my daily bread (Mishlei 30:8), and may I and all my household be provided with sustenance even before we ask, with favor and not with trouble, with permission and not with prohibition, with honor and not with disgrace, for life and for peace, from the stream of blessing and success, and from the supernal streaming wellspring, in order that I should be able to do Your will and be occupied with Your Torah and fulfill Your mitzvot.

May we not require handouts from flesh-and-blood. May You fulfill in me that is written in Tanach (Ps 145:16), "You open Your hand, and satisfy the desire of every living thing," and what is written (Ps 55:23), "Cast your burden upon ha-Shem, the One who shall sustain you."