My polygyny application was rejected. How should I end things with the woman I wanted to marry?

End it clearly, honestly and promptly; leaving her hanging hurts more than being straightforward. She too has built up hopes, and her family may already know. Tell her yourself about the court's decision and your decision from here. If you will not go ahead, say so, for example, ‘I don't want to leave you waiting for something uncertain.’ Do not quietly continue the relationship while promising to try again. Apologise to her family courteously, then focus fully on your wife and children. The Prophet ﷺ said: “A Muslim is the one from whose tongue and hand other Muslims are safe” (Sahih al-Bukhari 10). See guide for a prospective second wife.