Let $\pi(x)$ be the prime counting function and $P$ be the set of primes. Is $\Psi(k)\in P$ when the set of all $k$ consists of primes or semiprimes?
$$\Psi(k)=\int_0^k\pi(x)\pi(k-x)dx,$$
where $k\ge13.$
For example, for $k \le 540$, $\Psi(k)$ is prime for $15$ values, namely:
$k=\{13,57,119,167,171,173,175,341,395,397,427,431,473,515,519\}.$
If you are asking if the following implication holds: $$k \text{ is prime or semiprime} \implies \Psi(k) \text{ is prime}$$ then the answer is no: $k=19$ is prime but $\Psi(k) = 214$ is not.
If you are asking if the other implication holds: $$\Psi(k) \text{ is prime}\implies k \text{ is prime or semiprime} $$ then the answer is also no: $k = 171$ gives $\Psi(k) = 63097$ which is prime, but $k$ is not prime or semiprime as $171 = 3^2 * 9$.
If you are asking if this implication holds: $$\Psi(k) \text{ is prime}\implies k \text{ has at most 2 distinct prime factors} $$ then the answer is also no: $k = 585$ gives $\Psi(k) = 1530913$ which is prime, but $k$ is has 3 distinct prime factors $585 = 3^2 * 5 * 13$.