Given a polynomial f(x₁, x₂, ..., x_t) in t variables with integer coefficients and a positive integer n, we define α(n) as the number of integers 0 ≤ a < n such that the congruence f(x₁, x₂, ..., x_t) ≡ a (mod n) is solvable. We improve some known results for computing α(pⁿ), where p is prime and n ≥ 1, for polynomials of the form c₁x₁^k + c₂x₂^k + ··· + c_tx_t^k. We apply these results to calculate α(pⁿ) for polynomials of the form x^k ± y^k and to study the modular Waring problem.