  We study the existence of Khovanskii-finite valuations for rational curves of
arithmetic genus two. We provide a semi-explicit description of the locus of
degree $n+2$ rational curves in $\mathbb{P}^n$ of arithmetic genus two that
admit a Khovanskii-finite valuation. Furthermore, we describe an effective
method for determining if a rational curve of arithmetic genus two defined over
a number field admits a Khovanskii-finite valuation. This provides a criterion
for deciding if such curves admit a toric degeneration. Finally, we show that
rational curves with a single unibranched singularity are always
Khovanskii-finite if their arithmetic genus is sufficiently small.
