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+n<sup>2</sup> 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)<sup>2</sup> - n<sup>2</sup> will be bigger than p.