We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, $\sigma_1(\cdot)$,$\sigma_2(\cdot),\dots,\sigma_t(\cdot)$, of the primes such that the sum of vectors $$(p_1^2,p_2^2,\dots,p_k^2)+(p_{\sigma_1(1)}^2,p_{\sigma_1(2)}^2,\dots,p_{\sigma_1(k)}^2)+\dots+(p_{\sigma_t(1)}^2,p_{\sigma_t(2)}^2,\dots,p_{\sigma_t(k)}^2)=(T,T,T,....,T)$$ where $T=O(k^c)$ for some constant $c > 0$.

Is this possible and how do you do this? Given a $k$, can $t$ be as small as $O(\log(k))$?

For every $k$ is there a polynomially big $T$ and a $t$ that is logarithmic in $k$?

From Gerry's comments: let us fix $t=2$, for every $k$, is there a $T$ that is polynomial in $k$ that satisfies the above relations? His comments provide existence of values of $k$ such that $T=k$ but does not cover all $k$.