For what did Howard Hughes originally design the Zero fighter?

By all accounts, Hughes did not actually design the Zero fighter.

In his own words, "Now regarding the Japanese Zero... The Japanese Zero was a shock of the utmost magnitude to the United States because it had been thought up to that time that the Japanese were far inferior mechanically, I should say in point of aircraft design and mechanical aptitude, to the United States and nobody expected the Japanese to have an airplane that would be at all competitive. Well, in any event, when one of these Japanese Zeros was finally captured and studied and analyzed it was quite apparent to everyone that it had been copied from the Hughes plane which has been discussed earlier here. That is the only relationship between the Japanese Zero and the Hughes H-I design. I had no dealings with the Japanese or any other foreign government for the plane and to the best of everyone's knowledge the Japanese had no other access to it except through whatever espionage they may have had or through seeing photographs of it which naturally were published all over the world."

http://theaviatorhh.com/h1-racer.htm

See also here:
Notably the section on Hughes' H-1 influence

https://en.wikipedia.org/wiki/User_talk:Cglassey

(unlessI am misreading your question)