  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.
