Given a representative $\alpha$ for $[\alpha] \in \pi_q(X)$ we may attach a cell along $\alpha$ to form a new space $C(\alpha)$ (This really is the mapping cone). One would hope that this has the effect of killing off $\alpha$, that is, $$\pi_q(C(\alpha)) \cong \pi_q(X)/\langle \alpha \rangle.$$ In the case $q=1$ this follows quickly from the Seifert–van Kampen Theorem, since then one has \begin{align*} \pi_1(C(\alpha)) &\cong \pi_1(X) *_{\pi_1S^1} \pi_1(D^2)\\ &\cong \pi_1(X) *_{\pi_1S^1} *\\ &\cong \pi_1(X) / \langle \alpha \rangle,\\ \end{align*} as the amalgamation relates $\alpha$ to the trivial map. One of the comments points out this is not true in general for $q \ge 2$.
What is the analogous result for $q \ge 2$ (with references please)?