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.