Tropical trigonal curves: the general case

This paper is a follow-up of a previous work in which we show that, for a $3$-edge connected tropical curve $\Gamma$, the existence of a divisor of degree $3$ and Baker-Norine rank at least $1$ in $\Gamma$ is equivalent to the existence of a non-degenerate harmonic morphism of degree $3$ from a tropical modification of $\Gamma$ to a tropical rational curve. In this work, we extend this result to a tropical curve with lower edge connectivity which does not contain a cycle of (at least three) separating vertices (a so-called necklace).