Sums of one prime power and two squares of primes in short intervals

Abstract: Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{{k}+p_{2}{2}+p_{3}{2}$,} where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones previously known.