Would someone tell me whether my proof is correct or is it missing anything important, please?
Prove that if $\phi: G\to H$ is a homomorphism and $G_{1}\leq G$ is cyclic, then $\phi (G_{1})$ is cyclic.
Let $G_{1}$ be cyclic. Then $G_{1}=\left \langle g \right \rangle$ for some $g\in G_{1}$. Since $\phi(G_{1})=\left \{ \phi(g_{1}):g_{1}\in G_{1} \right \}$ and $g_{1}=g^{k}$ for all $g_{1}\in G_{1}$ and $k\in \mathbb{Z}$, by definition of homomorphisms, $\phi(g_{1})=\phi(g^{k})=(\phi(g))^{k}$ for all $\phi(g_{1})\in\phi(G_{1})$. Then $\phi(G_{1})=\left \langle \phi(g) \right \rangle$ hence cyclic.
Your proof is correct. Well done.
The phrasing at the end, though, is a little off; try a new sentence reading "Hence $\phi(G_1)$ is cyclic." It's just a minor suggestion.