This question is mainly inspired from a different problem I was working on.

Is there a value of $k$ such that, for each $n\in \mathbb N$, the equation $$\sum_{i=1}^{k}x_i^2=n$$ is solvable in $x_1,\dots ,x_k\in \mathbb P\cup \{0,1\}$? Here $\mathbb{P}$ is the set of prime numbers.

That is, can every nonnegative integer be written as a sum of squares of primes (together with $0$ and $1$), where the number of summands is absolutely bounded?

What about the same question but for sums of $n$th powers?

I want to know the research that has been done in this field.