For the positive odd integers $3\le k\le 25$, the equation $$p^2+q^2+r^2=3^k$$ with primes $p,q,r$ is solvable. Here is one solution for every exponent , calculated with PARI/GP :
[3, 3, 3, 3]
[5, 7, 13, 5]
[7, 17, 43, 7]
[5, 83, 113, 9]
[23, 173, 383, 11]
[109, 859, 919, 13]
[7, 677, 3727, 15]
[5, 2053, 11177, 17]
[5, 5659, 33619, 19]
[29, 35999, 95731, 21]
[277, 47407, 303143, 23]
[19, 124231, 912061, 25]
Does such a solution exist for every odd positive integer $k\ge 3\ $ ?
We can assume $p\le q\le r$, equality is allowed. Since $3^k\equiv 3\mod 8$ , a solution , if it exists , contains only odd primes and because of the divisibility by $3$, we can rule out prime $3$ as well except in the case $k=3$ ; either every summand or none must be disisible by $3$.
It is clear that evey power of $3$ is sum of three perfect squares, but it is unclear whether each square can be the square of a prime number.
I found multiple solutions for small exponents $k$, so there might be a solution for every $k\ge 3\ $.