Let $X_m$ be a space obtained from $S^1$ by attaching $D^2$ through the map $f(z)=z^m$ around the boundary. I have computed the homology group of it by exact sequence
$$\mathbb{Z} \cong H_1(S^1)\xrightarrow{\times m} H_1(X_m) \rightarrow H_1(X_m,S^1)\cong \widetilde{H}_1(X_m/S^1)\cong H_1(X_m/S^1)\cong H_1(S^2)\cong 0. $$
In particular, $$\mathbb{Z}\xrightarrow{\times m} H_1(X_m) \rightarrow 0$$
Thus, the first map is surjective so $H_1(X_m)\cong m\mathbb{Z}$.
Am I correct?
One can represent $X_m$ as a CW-complex with one $0$-cell, one $1$-cell and one $2$-cell. One gets a cellular chain complex $$0\to C_2\to C_1\to C_0\to0$$ where each $C_i\cong\Bbb Z$. The differential $C_1\to C_0$ is zero, so its kernel is $C_1$. The differential $C_2\to C_1$ takes a generator of $C_2$ to $m$ times a generator of $C_1$. Then $H_1(X_m)$ is the homology of this complex at the middle term, which is isomorphic to $\Bbb Z/m\Bbb Z$.