I have a (geometrically irreducible) cubic surface defined over a finite field $F_q$ with three non-$F_q$-rational singularities (defined over the cubic extension of $F_q$).

Counting the number of $F_{q^3}$-rational points is easy by projection from a singular point.

What can be said about the number of $F_q$-rational points of the surface?

I would like upper/lower bounds that are finer than the Lang-Weil bound.

Thanks, H.