Let $B \subset \mathbb{Z}^+$. Define $r_{B,h}(n)$ to be the number of ways of writing $n$ as $h$ elements of $B$ and $R_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT elements of $B$. In many applications, such as the Erdos-Tetalli theorem which finds a set $B$ such that $R_{B,h}(n) = \Theta(\log(n))$ it is more convenient to work with $R_{B,h}(n)$. The reason why such convenience does not affect the results is because in general, the number of elements in $B^h$ that sum to $n$ counted by $r_{B,h}(n)$ but not by $R_{B,h}(n)$, namely those with repeat summands, is negligible.

To illustrate with a fairly concrete example, consider the famous Goldbach Conjecture which asserts that every positive even integer larger than 2 can be written as the sum of two primes. In other words if $B$ is the set of primes, then Goldbach Conjecture is the assertion that $r_{B,2}(2n) > 0$ for all $n > 1$. But the truth is that one expects $r_{B,2}(2n)$ to tend to infinity. In this case the number of sums with repeat summands are precisely to write $2p = p + p$ for some prime $p$, and if $r_{B,2}(2n)$ does indeed tend to infinity, then this is a complete triviality to replace $r_{B,2}(2n)$ with $R_{B,2}(2n)$, since $0 \leq r_{B,2}(2n) - R_{B,2}(2n) \leq 1$ for all $n$.

So my question is, is there some general argument for this observation? That is, in a sufficiently general setting, one can essentially assume that the summands are distinct.

I note that in some very trivial cases this assumption is not appropriate at all. For example, consider $B =$ {$3k : k \in \mathbb{N}$} $\cup$ {$0,1$}. Then $B$ is an asymptotic basis of order 3, but it would NOT be a basis at all if we demanded that the summands be unique. This of course is a somewhat contrived example, since for a set of positive density the number of representations is very small. So of course our criteria for a 'sufficiently general setting' would have to exclude such trivial cases.