Is it permissible to supplicate using the mentioned formula to request the reunion with the one the supplicant loves and to facilitate other matters?

There is no objection to supplicating for the facilitation of a specific need. However, it is better for the believer to ask for the good of this world and the Hereafter through comprehensive supplications, such as: "Our Lord, grant us from among our wives and offspring comfort to our eyes and make us leaders of the righteous," because specifying a particular thing may not contain the hidden good. Allah Almighty said: "But perhaps you hate a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows, while you know not." A person may supplicate for evil, thinking it is good, due to their haste and ignorance of the consequences. Therefore, not specifying is more appropriate, and the supplicant should be content with Allah's decree in all circumstances.