Points on singular Frobenius nonclassical curves

In 1990, Hefez and Voloch proved that the number of $F_q$-rational points on a nonsingular plane $q$-Frobenius nonclassical curve of degree $d$ is $N = d(q-d+2)$. We address these curves in the singular setting. In particular, we prove that $d(q-d+2)$ is a lower bound on the number of $F_q$-rational points on such curves of degree $d$.