Let \(H\) be a semisimple Hopf algebra, finite dimensional over its splitting field \(K\), and let \(A\) be an \(H\)-module algebra. Assume that the smash product \(A\# H\) is semiprime. Then it is shown here that the Connes spectrum \(\text{CS}(A,H)\) is full, that is consists of all irreducible \(H\)-modules, if and only if every nonzero annihilator ideal of \(A\# H\) meets \(A\) nontrivially. Furthermore, if \(H\) is cocommutative, then this paper studies the intersection \(I\) of the annihilators of the irreducible modules in \(\text{CS}(A,H)\). Specifically, it considers the Hopf kernel of the natural map from \(H\) to \(H/I\).