The following result:
Lemma: There exist infinitely many prime numbers $p≥2$ and there exist positive integers $k,m$ such that $k^2+1=p+m$ with $1≤m≤p-1$.
is true as shown in There exist infinitely many prime numbers $p≥2$ and there exist positive integers $k,m$ such that $k^2+1=p+m$.
Now, consider an irreducible integer-valued polynomials $f(x)$ with integer coefficients.
Then I am asking about the necessary condition or possible forms of polynomials $f(x)$ verifying the following result:
There exist infinitely many prime numbers $p≥2$ and there exist positive integers $k,m$ such that $f(k)=p+m$ with $1≤m≤p-1$.