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 something finer than the Lang-Weil bound. 

Thanks,
  H.