[caldrail]

The Hughes racer was a private venture of Howard Hughes, and was never used by the US Air Force.

I know he never flew in the US Air Force, but you should know that the Bristol Blenhein also was an initiative of the private sector!
And I would like to know if there was some point in the interest of the US Air Force for this project

From Wikipedia....

Considering the contemporary service aircraft were biplanes, Hughes fully expected the United States Army Air Forces to embrace his aircraft's new design and make the H-1 the basis for a new generation of U.S. fighter aircraft. His efforts to "sell" the design were unsuccessful. In postwar testimony before the Senate, Hughes indicated that resistance to the innovative design was the basis for the USAAF rejection of the H-1: "I tried to sell that airplane to the Army but they turned it down because at that time the Army did not think a cantilever monoplane was proper for a pursuit ship..."

Aviation historians have posited that the H-1 Racer may have inspired later radial engine fighters such as the P-47 Thunderbolt and the Focke-Wulf Fw 190. After the war, Hughes further claimed that "it was quite apparent to everyone that the Japanese Zero fighter had been copied from the Hughes H-1 Racer." He noted both the wing planform, the tail empennage design and the general similarity of the Zero and his racer. Jiro Horikoshi, designer of the Mitsubishi Zero strongly refuted the allegation of the Hughes H-1 influencing the design of the Japanese fighter aircraft.