All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
It is not correct to count the Sunnah of the Fajr prayer if it is prayed after sunrise as the two rak'ahs of Ishraaq because each prayer is a Sunnah that is intended in itself, and the Sunan that are intended in themselves cannot be merged together.
The Fiqh Encyclopedia reads:
“If one joins two acts of worship with one intention, then if they are mergeable, such as both the ghusl for the Friday prayer and the ghusl for Janaabah (sexual defilement), or the ghusl for Janaabah and the ghusl for menstruation; or the ghusl for the Friday prayer and the ghusl for 'eed prayer, or that, on the other hand. one of them is not intended in itself, such as the prayer of salutation to the mosque when merged with an obligatory prayer, or another Sunnah, then this does not undermine the act of worship because purity (Ghusl) can be merged, and the prayer for the salutation to the mosque and similar prayers are not intended in themselves; rather, what is intended is occupying the place with prayer, so it may be included in another. As regards having a combined intention for two acts of worship both of which are intended in themselves, such as the Thuhr prayer and its Sunnah prayer, then it is not valid to join them in one single intention as they are two different acts of worship, and one of them may not be included in the other.”
Shaykh Ibn ‘Uthaymeen said:
“A person has missed the Sunnah of the Fajr until the sun has risen, and he came at the time of the Dhuha prayer, so here the Sunnah of the Fajr is not sufficient for the Dhuha prayer, and the Dhuha prayer is not sufficient for the Sunnah of the Fajr, and it is not sufficient to make one intention for both, because the Sunnah of the Fajr is independent and the Dhuha prayer is independent, so one of them is not sufficient for the other.”
He was also asked: “Can optional prayers be merged? For example: if I perform three rak’ahs as Witr prayer and I was lazy in performing them or I forgot to perform them, then I sat down after the Fajr prayer until the sun has risen, should I pray these two rak'ahs or should I pray four rak'ahs and intend them both for the Witr and for the Sunnah of the sunrise (Dhuha) altogether?" He replied: “They cannot be merged: the Sunnah of the Dhuha prayer is for the Dhuha prayer, and the Sunnah of Witr is for the Witr prayer, and the Sunnah Raatibah (i.e. the supererogatory prayer that is performed either before or after the obligatory prayer) is for the Sunnah Raatibah.”
Allaah knows best.