EDIT - I've rewritten my previous answer in an attempt to remove everything except the answer to your question. All submanifolds are assumed to be smooth.
Suppose that $M$ is a closed, connected, oriented, irreducible three-manifold (and $M$ is not the three-sphere). Suppose that $K$ is a knot in $M$. Then $\pi_2(M - K)$ is non-trivial if and only if $K$ is contained in an embedded three-ball $B^3 \subset M$.
The proof is an exercise using Alexander's theorem (every embedded two-sphere in $S^3$ bounds balls on both sides) and the sphere theorem. Hempel's book and Hatcher's notes on three-manifolds are standard references for the necessary background material.