Probability that a low-height sumset of early generators contains a prime

Determine, in the Erdős–Rényi-type random numerical semigroup model (p), the probability that the m-fold sumset G of the first k selected generators (conditioned to lie in {1, …, 2k/p}), with m exceeding log log(1/p), contains at least one prime number; quantify this probability as a function of p and the parameters k and m.

Background

To potentially remove extra logarithmic factors from their upper bounds, the authors propose replacing a large random prime generator by a prime drawn from a small sumset built from the first few generators selected by the ER-type process. They argue that if primes behaved like a random set of density 1/ln n, such a sumset would be expected to contain a prime.

A key missing step for this approach is to rigorously estimate the probability that this sumset indeed contains a small prime; resolving this would enable a refined bound and possibly reduce the logarithmic overhead.