I'm wondering if analytic number theorists can prove results which have the following flavor:

So let $N$ be a large positive integer.

Q: Can you always find a prime number $p$ in the interval $(N, 3N/2)$ for which there exists an odd prime $q$ which divides $p-N$ with multiplicity exactly one?

If such a result can be found in the literature I would like to have a reference. I have just not the single idea about where to start in order to prove such a result.

I kind of remember vaguely that every large enough even integer $N$ can be written as $p_1+p_2p_3$ where the $p_i$'s are prime numbers which is not that far from what I'm asking for.

pin arithmetic progressions (say with modulus the square ofq)? This is much easier than approximations to Goldbach, unless the intervals are very short. – Charles Matthews Mar 20 '11 at 22:57