If $m = 1$, then $m + \sum_{k=1}^n x^k/k!$ has no rational root for $n > 1$. And clearly the polynomial has a rational foot for all integers $m$ if $n = 1$. So, besides those cases, for what integers $m$ and positive integers $n > 1$ does $m + \sum_{k=1}^n x^k/k!$ have a rational root?
I'd be fine with a somewhat vague characterization as long as it captures all of the cases and leads to a decision procedure. E.g. something like $m,n$ satisfy $m = -\sum_{k=1}^n x^k/k!$ where $x$ is an integer such that $k!$ divides $x^k$ for all $k$ such that $1 \leq k \leq n$.