Is it permissible for a girl to supplicate that Allah unite her with someone she loves, and to complete the recitation of the Holy Quran with the intention of marrying him and gaining his affection? And is there a specific supplication for that?

It is permissible to supplicate for someone of sound religion and character, asking that marriage to them be facilitated, and that love for them be cast into one's heart.

There is no harm in reciting the Quran and interceding through it to achieve one's desires, as intercession through good deeds is legislated.

Thinking about worldly matters during Quran recitation contradicts the required contemplation, for the Prophet, peace and blessings be upon him, said: "Recite the Quran as long as your hearts are in agreement over it, but if you differ over it, then stand up (stop)."

There is no specific supplication for this purpose, but a Muslim should supplicate with whatever is easy for them from the supplications of success and facilitation of affairs.

It is recommended to perform Istikhara (prayer for guidance) when a suitor proposes, so that Allah may choose what is best.