Partial answer: Set a=1, so you want to enumerate the primes of the form b+c+bc = (b+1)(c+1)-1. This covers all primes p such that p+1 is a product of two factors of size at least 3. The leftovers (almost certainly covered by other values of a for sufficiently large primes) come from Sophie Germain pairs, which are conjectured to be infinite, but rather sparse.
More generally, the counterexamples are exactly those primes p such that for any positive n, p+n2 is not a product of two numbers strictly larger than n+1. It suffices to check for n up to p/4, since for larger n, (n+2)2 - n2 will be bigger than p.