||We study a subclass of p-ary functions in n variables, denoted by A(n), which is a collection of p-ary functions in n variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all p-ary (n - 1)-plateaued functions in n variables by proving that every (n - 1)-plateaued function should be contained in A(n). Secondly, we prove that if f is a p-ary r-plateaued function contained in A(n) with deg f > 1+n-r/4(p - 1), then the highest degree term of f is only a single term. Furthermore, we prove that there is no p-ary r-plateaued function in A(n) with maximum degree (p - 1)n-r/2+1. As application, we partially classify all (n - 2)-plateaued functions in A(n) when p = 3, 5, and 7, and p-ary bent functions in A(2) are completely classified for the cases p = 3 and 5.