Let $h \in \mathbb{C}[x]$ with $\deg(h)=d \geq 1$ and let $R_{h,y}:=\mathbb{C}+\langle h,y \rangle \subseteq \mathbb{C}[x,y]$.
It is known that subalgebras of $\mathbb{C}[x]$ are finitely generated, see this, therefore, $\mathbb{C}+\langle h \rangle$ is finitely generated, and it is generated by $\{h,xh,\ldots,x^{d-1}h\}$. For example, for $h=x^2$ we obtain $\mathbb{C}+\langle x^2 \rangle=\mathbb{C}[x^2,x^3]$, so a finite set of generators is, for example $\{x^2,x^3\}$.
There are non-finitely generated subalgebras of $\mathbb{C}[x,y]$, see this.
Is $R_h$ finitely generated? Probably not? It is not true that $R_{h,y}=\mathbb{C}[y, x^iy^jh]_{0 \leq i+j \leq d-1}$, since $xy \in \mathbb{C}+\langle h,y \rangle$, but $xy \notin \mathbb{C}[y, x^iy^jh]_{0 \leq i+j \leq d-1}$.
Edit: Could one please present a finite list of generators, if exists.
Thank you very much!