Plane $\mathbb{A}^1$-curves on the complement of strange rational curves

A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in characteristic $p$. We prove the connectedness of the moduli spaces of $\mathbb{A}^1$-curves with given degree, classify their irreducible components, and exhibit the inseparable $\mathbb{A}^1$-connectedness via the $\mathbb{A}^1$-curves parameterized by each irreducible component. The key to these results is the strangeness of all $\mathbb{A}^1$-curves. As an application, in every characteristic we construct explicit covering families of $\mathbb{A}^1$-curves, whose total spaces are smooth along large numbers of cusps on each general fiber.