The Sombor index of a graph $G$ was recently introduced by Gutman from the geometric point of view, defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{d(u)^2+d(v)^2}$, where $d(u)$ is the degree of a vertex $u$. For two real numbers $α$ and $β$, the $α$-Sombor index and general Sombor index of $G$ are two generalized forms of the Sombor index defined as $SO_α(G)=\sum_{uv\in E(G)}(d(u)^α+d(v)^α)^{1/α}$ and $SO_α(G;β)=\sum_{uv\in E(G)}(d(u)^α+d(v)^α)^β$, respectively. A $k$-polygonal cactus is a connected graph in which every block is a cycle of length $k$. In this paper, we establish a lower bound on $α$-Sombor index for $k$-polygonal cacti and show that the bound is attained only by chemical $k$-polygonal cacti. The extremal $k$-polygonal cacti for $SO_α(G;β)$ with some particular $α$ and $β$ are also considered.