In the letter, he says he realizes that he was wrong about Jane not loving Bingley, but he was worried about how everyone in the family was acting. (Except Jane and Lizzy, natch.)
He actually feels okay about what he did, except for the part where he didn't tell Bingley that Jane was in London.
As for Wickham, the story is actually totally different: Darcy gave Wickham the 1000 pounds that the elder Mr. Darcy left, and promised him the living eventually—but Wickham wanted to study law instead, and gave up claim to the living in exchange for 3000 pounds.
A few years later, Wickham showed up with no money, no law degree, a really bad reputation, and a burning desire to be ordained, i.e. become a clergyman—which Darcy refused.
But Wickham had a plan B: he wormed his way into the heart of Darcy's 15-year-old little sister Georgiana and tried to elope with her. Darcy found out just in time.