If $e$ is "small" (as all typical values of $e$ are!), this is vulnerable to the Franklin-Reiter related message attack.
In a nutshell, given $c_1=x^e\bmod n$ and $c_2=(x+m)^e\bmod n$, for each possible value of $m$, compute
in the polynomial ring $(\mathbb Z/n)[y]$. If you guessed the right $m$, then (possibly except for very rare cases)
and you have recovered $x$ and verified that $m$ was correct. Otherwise, $f$ will most likely be $1$. Note that this $\gcd$ is technically not well-defined in all cases, but if you actually encounter a problem with that, you have effectively factored $n$.
Of course, this assumes that $e$ is small enough for this $\gcd$ computation to be feasible, and that the set of possible messages $m$ is small enough to be brute-forceable.