Is there an integer $n$ with an infinite number of representations of the form $n=2q-p$, where $p$ and $q$ are both primes?

Given a positive integer $k>1$, I would like to know for which (if any) integers $n$ the linear equation $q-kp=n$ admits an infinite number of solutions, where $p$ and $q$ are primes. (I'm not including $k=1$ because it reduces to well know open problems, $k=1$ and $n=2$ would be the twin primes conjecture)

The density of the prime numbers implies that at least there are integers $n$ with an arbitrarily large number of representations.