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The adjunction formula on the degree-$d$ projective hypersurface gives $K_X=(d-4)H$. Since $C$ is a smooth rational curve, adjunction on $C$ gives $(K_X+C)\cdot C=2g(C)-2=-2$. Its degree is $H\cdot C=a$, so the smooth rational-curve self-intersection on a projective hypersurface formula is

The Noether–Lefschetz theorem makes any curve class a positive multiple of the hyperplane class, with positive self-intersection. But the smooth rational-curve self-intersection on a projective hypersurface formula makes every smooth rational curve have self-intersection at most $-2$ when the surface degree is at least four. Thus none exists. The smoothness condition is essential; this does not prohibit singular curves whose normalizations are rational.