= Solution
Let $q:S^2\to X$ identify the two points to $p$. Take disjoint small open disks around the original points and let $U$ be their image; $U$ is a contractible wedge of two disks. Let $V=X\setminus\{p\}$, which is homeomorphic to a sphere with two punctures and retracts onto a circle. Then $U\cap V$ consists of two punctured disks, each retracting onto a circle.
In the <Mayer–Vietoris sequence>, the map $H_1(U\cap V)=\mathbb Z^2\to H_1(U)\oplus H_1(V)=\mathbb Z$ has rank one and is surjective: both annular loops map to generators, with a possible orientation sign. Its kernel is $\mathbb Z$, so $H_2(X)=\mathbb Z$. At degree zero the map $\mathbb Z^2\to\mathbb Z\oplus\mathbb Z$ is $(a,b)\mapsto(a+b,-a-b)$, with rank-one kernel. The preceding degree-one map is surjective, so exactness gives $H_1(X)=\mathbb Z$. Since $X$ is connected and both pieces have no higher homology,
$$
\boxed{H_j(X;\mathbb Z)=\begin{cases}\mathbb Z,&j=0,1,2,\\0,&j\geq3.\end{cases}\qquad b_0=b_1=b_2=1.}
$$
This illustrates <homology after identifying two points on a sphere>.
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