In this problem, $m$ and $n$ are integers.

If $m\equiv 4\pmod{14}$ and $m-n\equiv 10\pmod{14}$, then what is the remainder when $m+n$ is divided by $14$?
We have \begin{align*}
m+n &= 2m - (m-n) \\
&\equiv 8 - 10 \\
&\equiv -2 \\
&\equiv -2+14 \\
&\equiv{12} \pmod{14},
\end{align*}so the remainder is $\boxed{12}$.