Is reciting Surat Al-Mulk after Fajr prayer a reason for conception, and is there a specific supplication for seeking offspring?

Designating Surat Al-Mulk to be recited after Fajr prayer with the intention of conceiving is an innovation (bid'ah) for which there is no evidence. The established virtue of Surat Al-Mulk is that it saves one from the torment of the grave for whoever regularly recites it. Any pregnancy that may occur after reciting it is a mere coincidence and by Allah's decree, and not an evidence for the validity of designating this Surah for this purpose.

There is no specific supplication for conception, but supplication in general and asking for righteous offspring is beneficial. Among the best supplications mentioned in the Quran is the supplication of Zakariya (peace be upon him): (My Lord, grant me from Yourself a good offspring. Indeed, You are the Hearer of supplication).