Suppose, $n\ge 24$ is a natual number. Then, $n$ can be written as a sum of squares of primes (the squares of the primes need not be distinct) because $n$ can always be written in the form $4a+9b$ with non-negative integers $a,b$. So, we can find primes (not necessarily distinct) such that $$(1)\ \ \ \ p_1^2+p_2^2+\cdots+p_k^2=n$$
But is there a positive integer $k$, such that every positive integer $n\ge16$ can be written as a sum of at most $k$ squares of a prime ? In other words, a number $k$ , such that $(1)$ has always a solution ?
This is known as the "Waring-Goldbach problem". Waring's problem is of course your problem without the prime constraint, and Goldbach's problem is your problem for first powers. This article by Buttcane states that at least seven, and no more than nine, squares of primes are necessary to express every sufficiently large integer.