It is known that if every prime power branched cyclic cover of a knot in $S^3$ is a homology sphere, then the knot has vanishing Casson–Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in $S^3$ whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of $\scrf_{(1.0)}/\scrf_{(1.5)}$ for which Casson–Gordon invariants vanish in Cochran–Orr–Teichner's filtration of the classical knot concordance group. As a corollary, it follows that Casson–Gordon invariants are not a complete set of obstructions to a second layer of Whitney disks.