= Solution
Let $T$ be the <surface with boundary> obtained by deleting the interiors of the $n$ disks. By <excision>,
$$
H_i(S,T)\cong\bigoplus_{j=1}^nH_i(D^2,S^1),
$$
so $H_2(S,T)\cong\mathbb Z^n$ and the other relative groups vanish. In the <long exact sequence in relative homology>, the map
$$
H_2(S)\cong\mathbb Z\longrightarrow H_2(S,T)\cong\mathbb Z^n
$$
sends an <orientation class> to $(1,\ldots,1)$, up to an overall choice of sign. It is therefore injective and has <cokernel> $\mathbb Z^{n-1}$. Exactness gives $H_2(T)=0$ and a <short exact sequence>
$$
0\longrightarrow\mathbb Z^{n-1}\longrightarrow H_1(T)\longrightarrow H_1(S)\cong\mathbb Z^{2g}\longrightarrow0.
$$
This sequence splits because its right-hand group is a <free abelian group>. Since $T$ is connected, we obtain
$$
\boxed{H_i(T;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&i=0,\\
\mathbb Z^{2g+n-1},&i=1,\\
0,&\text{otherwise}.
\end{cases}}
$$
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