Statement (a) is true for $N$ sufficiently large. Specifically, choose $p_4=2$ for $N$ odd, and $p_4=3$ for $N$ even. Then it suffices to show that every sufficiently large odd number $M$ can be written as $p_1+p_2-p_3$ with odd primes $p_1,p_2,p_3$. Fix $M$, and consider the primes $M<p_3<2M$, then it suffices to show that among the numbers of the form $M+p_3$ there is one that can be written as a sum of two odd primes. We are talking about $\gg M/\log M$ even numbers in $(M,2M)$, whereas Vaughan (1972) proved that at most $M\exp(-c\sqrt{\log M})$ of them cannot be written as a sum of two primes (much better bounds are known now, thanks to Montgomery-Vaughan (1975) and refinements). So statement (a) is true for $N$ sufficiently large, and current technology is probably capable of establishing it for all $N$'s.

Statement (b) seems to be out of reach. Specializing to $b=7$, it states that either $a$ or $a-7$ is a difference of two primes (because $7$ is not such a difference). Specializing further that $a$ is odd but not of the form $p-2$ (with $p$ a prime), the conclusion is that $a-7$ is a difference of two primes. So (b) implies that almost all even numbers (namely all even numbers but the numbers $p-9$) is a difference of two primes. Currently we know by the recent breakthroughs around the twin prime conjecture (Zhang, Maynard, Tao, Polymath8) that a positive proportion of the even integers can be written as a difference of two primes (in fact the density of such even integers exceeds $1/354$), but almost all even integers seems to be out of reach. On the other hand, assuming a generalized [Elliott-Halberstam conjecture][1], statement (b) follows for any even integers $0<b<a$ such that $a$ and $b$ do not cover both nonzero residues modulo $3$ (i.e. if $a$ and $b$ are not divisible by $3$ then they have the same residue modulo $3$). Specifically, under these hypotheses, [Polymath8b][2] proved that infinitely many translates of $\{0,b,a\}$ contain at least two primes, hence in particular one of $a$, $b$ , $a-b$ is a difference of two primes infinitely often.

To summarize, (a) is essentially known, while (b) seems to be out of reach.


  [1]: https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture
  [2]: https://arxiv.org/abs/1407.4897