Waring's problem in natural numbers of special form

Let $\mathbb N_0$ be the set of positive integers whose binary decompositions contain an even number of ones. We give a bound for the trigonometric sum of special form over numbers in $\mathbb N_0$; using this bound, we derive an asymptotic formula for the number of solutions to Waring's equation in positive integers in $\mathbb N_0$, and also a bound for the number of terms in the last equation, which is sufficient for the equation to be solvable in integers in $\mathbb N_0$.
Bibliography: 9 titles.