Counting the solutions of ${\lambda }_1{x}_1^{k_1}+\cdots +{\lambda }_t{x}_t^{k_t}\equiv c<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>n$

Given a polynomial $Q(x_1,\cdots, x_t)=\lambda_1 x_1^{k_1}+\cdots +\lambda_t x_t^{k_t}$,forevery$c\in $\mathbb{Z}$$and$n\geq 2$, we study the number of solutions $N_J(Q;c,n)$ of the congruence equation $Q(x_1,\cdots, x_t)\equiv c\bmod{n}$in$($\mathbb{Z}$/n$\mathbb{Z}$)^t$suchthat$x_i\in ($\mathbb{Z}$/n$\mathbb{Z}$)^\times$for$i\in J\subseteq I= {1,\cdots, t}$. We deduce formulas and an algorithm to study $N_J(Q;c,p^a)$ for $p$ any prime number and $a\ge 1$ any integer. As consequences of our main results, we completely solve: the counting problem of $Q(x_i)=\sum\limits_{i\in I}\lambda_i x_i$foranyprime$p$andanysubset$J$of$I$;thecountingproblemof$Q(x_i)=\sum\limits_{i\in I}\lambda_i x^2_i$inthecase$t=2$forany$p$and$J$,andthecase$t$generalforany$p$and$J$satisfying$\min{v_p(\lambda_i)\mid i\in I}=\min{v_p(\lambda_i)\mid i\in J}$; the counting problem of $Q(x_i)=\sum _{i\in I}{\lambda }_i{x}_i^k$ in the case $t=2$ for any $p\nmid k$ and any $J$, and in the case $t$ general for any $p\nmid k$ and $J$ satisfying $\min{v_p(\lambda_i)\mid i\in I}=\min{v_p(\lambda_i)\mid i\in J}$.