Remarks on additive representations of natural numbers

For two relatively prime square-free positive integers $a$ and $b$, we study integers of the form $ap+b{P}_2$ and give a new lower bound for the number of such representations, where $ap$ and $b{P}_2$ are both square-free, $p$ denote a prime, and $P_2$ has at most two prime factors. We also consider some special cases where $p$ is small, $p$ and $P_2$ are within short intervals, $p$ and $P_2$ are within arithmetical progressions and a Goldbach-type upper bound result. Our new results generalize and improve previous results.