I am not convinced that the definition of $c_k$ makes sense.
Consider a simpler problem:
Since there are infinitely many primes we can define $g(n)=\min\{k \ge 0 \mid n+k \in \mathbb{P}\}.$
- Would it make sense to say $P(n+k \in \mathbb{P})=\frac{d_k}{\log n}?$
- At any rate, would you entertain the notion that $g(n)=O(\log n)?$
Actually it is known that is false. In fact, Cramer's conjecture is that $g(n)=O((\log n)^2).$ The heuristic supports the stronger claim $\limsup \frac{g(n)}{(\log n)^2}=1.$ Of course only prime $n$ need be considered.