= Solution
Write $X'=X\cup_A CA$. After replacing the two pieces by small open neighborhoods, their intersection deformation retracts to $A$. The <Mayer–Vietoris theorem>[Mayer-Vietoris sequence], together with $H_i(CA)=0$ for $i>0$ and $H_0(CA)=\mathbb Z$, becomes
$$
\cdots\to H_{i+1}(X')\to H_i(A)\to H_i(X)\to H_i(X')\to H_{i-1}(A)\to\cdots
$$
and at degree zero ends as
$$
\cdots\to H_1(X')\to H_0(A)\to\mathbb Z\oplus H_0(X)\to H_0(X')\to0,
$$
which is the required sequence.
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