On the number of solutions of certain diagonal equations over finite fields

We use character sums over finite fields to give formulas for the number of solutions of certain diagonal equations of the form$a_1{x}_1^{m_1}+a_2{x}_2^{m_2}+\cdots +a_n{x}_n^{m_n}=c.$ We also show that if the value distribution of character sums ${\sum }_{x\in {\mathbb{F}}_q}\chi (a{x}^m+bx)$, $a,b\in {\mathbb{F}}_q$, is known, then one can obtain the number of solutions of the system of equations$\{\begin{array}{cc}\hfill x_1+x_2+\cdots +x_n\hfill & \hfill =\alpha \hfill \\ \hfill x_1^m+x_2^m+\cdots +x_n^m\hfill & \hfill =\beta ,\hfill \end{array}$ for some particular m. We finally apply our results to induce some facts about Waring's problems and the covering radius of certain cyclic codes.