Can we recite the supplication of "Āyat al-Kursī" (the Throne Verse) to achieve our goals? What is the best time to recite it? What is its importance according to Ḥadīth?
The Throne Verse (Ayat-al-Kursī) is not a supplication rather it would be correct to render it the most comprehensive statement of the belief in one God. A believer declares his conviction that the Almighty is the sole owner of all the power and sovereignty and declares that he is pure of any attributes, which do not match his mighty place. When we recite this verse we in fact are declaring our limitations and hand the affairs to the Almighty. The wording of the declaration is from the Almighty himself hence the best. I don't believe that mere recitation of a verse to achieve some worldly goal is recommendable in Islam. Islam required us to be true believers and to live our life according to the directives of Sharī'ah and hand our affairs to God. It teaches us to pray to God for help in all the affairs. However the elements of charms which guarantee fulfillment of your requirements are nonexistent in Islamic religious sources.