It is worth to mention that in his original paper, E. Warning also find a lower bound for the number of solutions:
Apart from the references provided by other answers, an excellent survey about Chevalley-Waring theorem, its history and applications and related problems about solving polynomial equations over finite fields, written by Jean-René Joly, is accessible from
Équations et variétés algébriques sur un corps fini (L'Enseignement Mathématique (1973))