Translation:He is not singing, even though he wants to.
"He doesn't sing, despite that he wants to." was not accepted. Should it have been?
It does not sound right, and my English grammar is not quite strong enough to pinpoint exactly why.
I think it is that it uses the wrong form of "want". You could say: "despite wanting to", though this still sounds awkward. Definitely better to stick with "even though" in this sentence.