  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 $\cF$ 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.
