Answered by Shaykh Faraz Rabbani
In a fard prayer, I learned that it is necessary to not delay the wajib parts of the salah (Acc. to Sh. Yaqoubi). So, along those lines, I learned from various sources some of the adhkar and du’as that should be made at different times of the salah, such as when one is in sujud, or after the tashahhud, etc. But, a friend of mine raised the point that those prayers and adhkar are SUNNAH, and thus of lesser degree than the wajib and fard, so is it permissible, as far as you know, to do recite those things in the fard prayer, or should they be done in extra prayers because doing them may be delaying the wajib?
The detail of what Sayyidna Shaykh Yaqoubi, Allah preserve him, mentioned is that is it necessary (wajib) not to interrupt or delay the prayer with that which is not of it unnecessarily for the extent of more than an integral or the prayer (=the extent of 3 tasbihs).
Therefore, for example, if one stood day-dreaming after one’s recitation, without reciting, for more than the extent of three tasbihs (=saying Subhan Allah at normal pace), one will have left this wajib. Of course, things like coughing or pausing a moment in reflection of the meaning of the verses are not of this. [Ibn ‘Abidin, and Mufti Mahmoud Usmani’s personal explanation]
There is a specific wajib, related to this, which is to stand up from the tashahhud  after the first sitting of a 3 or 4 rakat prayer immediately. Here, an unnecessary delay of more than the time it takes to say “Allahummasalli`ala Muhammadin wa`ala Aali Muhammad” would entail leaving a wajib. [Ibn Abidin]
As regards to the supplications in ruku‘ and sujud, there is some detail regarding them:
1. What is a confirmed sunna in the bowing and prostration is their respective tasbihs,  three times. It is important not to do this less than three times because Ibn Abidin mentions that the principles of the Hanafi school and the evidence would imply that it is necessary (wajib) to do both tasbihs three times.
However, he notes, the transmitted position of the school is that doing so is a confirmed sunna, and leaving a confirmed sunna even once without excuse is blameworthy, though not quite sinful. Making it a habit is sinful, because it is considered leaving the inveterate way of the Prophet (Allah bless him and give him peace), who said, “Whoever leaves my way (sunna) is not of me.” That is, is not of those who follow my example fully.
2. Supplications in bowing and prostration are not a confirmed sunna. They are recommended, however, without disagreement, in supererogatory prayers. [Maraqi al-Falah, Ibn Abidin]
As for the obligatory prayer, it is disliked for the imam to supplicate while bowing or prostrating, after the confirmed sunna tasbihs, because the imam has been ordered not to elongate his prayer, and this is considered part of elongating one’s prayer. However, what Ibn Abidin quotes from Sharh al-Munyaindicates that it is not disliked for the follower or one praying the obligatory prayer alone to make these supplications.
This is what I learned from Shaykh Muhammad Jumu‘a, and have seen from the practice of leading Hanafi scholars whose knowledge I respect.
3. Note, also, that it is a confirmed sunna to supplicate in the final sitting after sending blessings on the Prophet (Allah bless him and give him peace) before the final salams. Any supplication is minimally sufficient, though it is best to choose a transmitted supplication.
If one supplicates from oneself, it should not be for worldly matters, for such supplications invalidate the prayer, because the Prophet (Allah bless him and give him peace) said, “Verily, no human speech befits this prayer of ours.” Worldly matters are things one can ask humans for, such as, “Allah marry me to so-and-so,” or, “Allah, I need a new Lexus.” [Shurunbulali, Maraqi al-Falah; Ibn ‘Abidin, Radd al-Muhtar]
And Allah knows best.
[Shaykh] Faraz Rabbani
Shaykh Faraz Rabbani spent ten years studying with some of the leading scholars of recent times, first in Damascus, and then in Amman, Jordan. His teachers include the foremost theologian of recent times in Damascus, the late Shaykh Adib al-Kallas (may Allah have mercy on him), as well as his student Shaykh Hassan al-Hindi, one of the leading Hanafi fuqaha of the present age. He returned to Canada in 2007, where he founded SeekersGuidance in order to meet the urgent need to spread Islamic knowledge–both online and on the ground–in a reliable, relevant, inspiring, and accessible manner. He is the author of: Absolute Essentials of Islam: Faith, Prayer, and the Path of Salvation According to the Hanafi School (White Thread Press, 2004.) Since 2011, Shaykh Faraz has been named one of the 500 most influential Muslims by the Royal Islamic Strategic Studies Center.
 The tashahhud is to sayat-Tahiyyatlillah… in the first and final sitting.
 This is to say:Subhana Rabbiy a’l`Adheem while bowing, and Subhana Rabbiy a’lA`la while prostrating.