Is there a specific manner for supplication during prayer?

The best guidance is the guidance of Muhammad, peace be upon him, and the best supplication is that which conforms to his Sunnah; the wordings are fixed and must be preserved.

Confirmed supplications from him, peace be upon him, in prayer:

1. After the opening takbeer (Dua al-Istiftah): "O Allah, distance me from my sins as You have distanced the East and the West. O Allah, cleanse me from my sins as a white garment is cleansed from dirt. O Allah, wash me from my sins with snow, water, and hail."

2. Dua al-Qunut in Witr: "O Allah, guide me among those You have guided, grant me well-being among those You have granted well-being, take me into Your care among those You have taken into Your care, and protect me from the evil of what You have decreed. For You decree and none can decree against You. Indeed, none whom You have befriended shall be humiliated, and none whom You have opposed shall be honored. Blessed and Exalted are You, our Lord."

3. During bowing (ruku') and prostration (sujud): "Glorified be You, O Allah, our Lord, and with Your praise, O Allah, forgive me." Supplication in prostration is better due to the servant's closeness to his Lord at that time.

4. Between the two prostrations: "O Allah, forgive me, have mercy on me, mend my affairs, guide me, and provide for me."

5. After the Tashahhud and before the Tasleem: "O Allah, I seek refuge in You from the punishment of Hell, and from the punishment of the grave, and from the trials of life and death, and from the evil of the trial of the Antichrist." "O Allah, I have greatly wronged myself, and none forgives sins except You. So grant me forgiveness from You and have mercy on me. Indeed, You are the Forgiving, the Merciful." When he, peace be upon him, stood for prayer, he would say: "I have turned my face sincerely towards the One who originated the heavens and the earth, and I am not of the polytheists... Blessed and Exalted are You, I seek Your forgiveness and repent to You." And when he bowed, he would say: "O Allah, to You I have bowed, in You I have believed, and to You I have submitted. My hearing, my sight, my mind, my bones, and my nerves are humbled before You." And when he raised his head, he would say: "O Allah, our Lord, to You be praise, filling the heavens, and filling the earth, and filling what is between them, and filling whatever else You wish." And when he prostrated, he would say: "O Allah, to You I have prostrated, in You I have believed, and to You I have submitted. My face has prostrated to the One who created it, fashioned it, and gave it hearing and sight. Blessed is Allah, the Best of creators." And the last thing he would say between the Tashahhud and the Tasleem: "O Allah, forgive me for what I have sent forth and what I have held back, what I have hidden and what I have revealed, what I have done in excess, and what You know better than I. You are the Foremost and You are the Last. There is no god but You."