analytic number theory - complete estimates of the error for a well-known asymptotic expression of partition p(n,m)

I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved here that
$$p(n,m)sim frac{n^{m-1}}{m!(m-1)!}$$ holds for $m=o(n^{1/3})$. In generality for any finite set $A$, with $|A|=m$ and $p(n,A)$ denoting the number of partitions of $n$ with parts from $A$, one has
$$p(n,A)=frac{1}{prod_{ain A}a}frac{n^{m-1}}{(m-1)!}+O(n^{m-2}).$$

Such estimations can be deduced from the generating function of $p$ by using methods that are described in many books, for example "Analytic Combinatorics" by Flajolet and Sedgewick.