Number of rational branches of a plane singular curve over a finite field

Let $\mathcal{F}$ be a plane singular curve defined over a finite field ${\mathbb{F}}_q$. The linear system of plane curves of a given degree passing through the singularities of $\text{cF}$ provides potentially good bounds for the number of points on a non-singular model of $\mathcal{F}$. In this note, the case of a curve with two singularities such that the sum of their multiplicities is precisely the degree of the curve is investigated in more depth. In particular, such plane models are completely characterized, and for $p>3$, a curve of this type attaining one of the obtained bounds is presented.