Let $(X,A)$ and $(Y,B)$ be CW pairs, $[X,A;Y,B]$ be the homotopy set of maps from $(X,A)$ to $(Y,B)$. Is there any conditions that guarantee the bijection induced by the quotient:$$[CX,X;Y,B]\longrightarrow [\Sigma X,Y/B]~~?$$ In particular, I am very interested in the case $X=M(\pi,n)$, the Moore space with the $H_n(X)=\pi$, where $\pi$ is an abelian group, finite or not.
A well-known result is Prop.4.28 of Hatcher's book "algebraic topology", which is a corollary of the Blakers-Massey Theorem.
If $X=S^{m-1}$, $(Y,B)$ is $r$-connected and $B$ is $s$-connected, then for any $m\leq r+s$, there holds an isomorphism: $$\pi_m(Y,B)\cong \pi_m(Y/B).$$
Is there a version of B-M theorem for homotopy groups with coefficients?