A curious pattern on primes congruent to $1$ mod $4$?

This is not true for $p=41$ because, except for permutations and signs, the only possible values are $a=5$ and $b=4$ but $a-b=1$ and $a+b=9$, not primes.

Another counterexample is $p=353$, for which $a=17$ and $b=8$ but $a-b=9$ and $a+b=25$, both squares!

Here are the first few counterexamples:

\begin{array}{rccccc} p & a & b & a-b & a+b \\ 41 & 5 & 4 & 1 & 9 \\ 113 & 8 & 7 & 1 & 15 \\ 313 & 13 & 12 & 1 & 25 \\ 353 & 17 & 8 & 9 & 25 \\ 613 & 18 & 17 & 1 & 35 \\ 653 & 22 & 13 & 9 & 35 \\ 677 & 26 & 1 & 25 & 27 \\ 761 & 20 & 19 & 1 & 39 \\ 857 & 29 & 4 & 25 & 33 \\ 977 & 31 & 4 & 27 & 35 \\ \end{array}

Let $a+b=35$ and $a-b=9$. Neither is prime.

Then $a=22$ and $b=13$, and the sum $(22)^2+(13)^2$ is the prime $653$.

Remark: For nice examples of apparent patterns that disappear when we look at larger numbers, please see Richard Guy's The Strong Law of Small numbers.