On the average number of representations of an integer as a sum of like prime powers

We investigate the average number of representations of a positive integer as the sum of $k+1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k=2$ [6] and $k = 3$ [5] respectively. We use the same technique to study the corresponding problem for sums of just $k$ perfect $k$-th powers of primes.