It is worth to mention that in his original paper, E. Warning also find a lower bound for the number of solutions:

![enter image description here][1]

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))][2] 


  [1]: https://i.sstatic.net/XP40k.jpg
  [2]: http://retro.seals.ch/digbib/view?pid=ensmat-001:1973:19::6