All perfect praise be to Allah, The Lord of The Worlds; and may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
Salatul Kasuf(Prayer of solar and lunar eclipse) is a confirmed Sunnah. Abu Baker(May Allah be pleased with him) narrated: “ We were with Allah's Apostle when the sun eclipsed. Allah's Apostle stood up dragging his cloak till he entered the Mosque. He led us in a two-Rakat prayer till the sun (eclipse) had cleared. Then the Prophet (p.b.u.h) said, "The sun and the moon do not eclipse because of someone's death. So whenever you see these eclipses pray and invoke (Allah) till the eclipse is over."{Bukhari&Muslim}.
The time for al-Kasuf-whether partial or complete-is from the beginning of the eclipse until the eclipse finishes for the Prophet(PBUH) said: “So when yo see them, invoke Allah and pray till the eclipse is clear.”{Bukhari}
It is best to make Ghusl (ceremonial washing) and pray it in congregation in the mosque without raising Azaan or Eqama; rather, it is called for by saying “ as-salatu jami'ah ”Prayer in congregation”. After the prayer, the Imam delivers two speeches (sermons) like in Eid prayer.
Praying it in congregation is better although it is permissible for a single person to perform it and for a woman to offer while in her house as stated in the book (Moghni Al-Mohtajj (599/1): “ It is permissible for a single person, a slave, a woman, and a traveler.”
This prayer consists of two Rakat, and that in every Rakat a person should perform two standings (Qiyam), two bowings (Ruku), and recite Al-Fatihah in each standing. The perfect form of eclipse prayer is: Make Takbir, recite Al-Istiftah supplication, seek refuge in Allah from the evil whispers of the devil, silent recitation of surat Al-Fatihah and surat Al-Baqarah, or a chapter of the same length, make a long bow and glorify (Tasbeeh) Allah for a period that equals reciting a hundred verses, stand up and glorify as well as thank Allah while in that position. Recite Al-Fatihah, a chapter shorter than the first one (Al-Baqarah): Al-Emran, or a chapter of the same length, make a long bow, but shorter than the first one, stand up and glorify Allah as well as give thanks to Him, but less than the first time, make two long Sajdahs (prostrations) without waiting for a long time between them, stand up and repeat the same acts offered in the first Rakat, but with shorter length, then offer Tashahud and Tasleem.
It is stated in (Umdat As-Salik, p.87): “ Its perfect form is: To recite after the Istiftah supplication, seeking refuge in Allah, and Al-Fatihah: Al-Baqarah in the first standing (Qiyam), Al-Emran in the second, An-Nissa` in the third, Al-Mai`da in the fourth or chapters of similar length, offer Tasbeeh (Glorification of Allah) in the first bow (Ruku) for a period parallel to the reciting of a hundred verses from chapter Al-Baqarah, parallel to eighty in the second, seventy in the third, fifty in the fourth, and continue the rest of this prayer as is the case with other prayers, then the Imam delivers two speeches (sermons) similar to Friday prayer.”
Abdullah Bin Abbas narrated: “During the lifetime of Allah's Apostle, the sun eclipsed. Allah's Apostle offered the prayer of (the) eclipse) and so did the people along with him. He performed a long Qiyam (standing posture) during which Surat-al-Baqara could have been recited; then he performed a pro-longed bowing, then raised his head and stood for a long time which was slightly less than that of the first Qiyam (and recited Qur'an). Then he performed a prolonged bowing again but the period was shorter than the period of the first bowing, then he stood up and then prostrated. Again he stood up, but this time the period of standing was less than the first standing. Then he performed a prolonged bowing but of a lesser duration than the first, then he stood up again for a long time but for a lesser duration than the first. Then he performed a prolonged bowing but of lesser duration than the first, and then he again stood up, and then prostrated and then finished his prayer. By then the sun eclipse had cleared. The Prophet then said, "The sun and the moon are two signs among the signs of Allah, and they do not eclipse because of the death or birth of someone, so when you observe the eclipse, remember Allah (offer the eclipse prayer).” {Bukhari&Muslim}
Finally, there is no preferred supplication to Al-Kasuf, but it is better to offer lots of Istigfar (seeking forgiveness), Takbir, charity, and righteous deeds. And Allah knows best.