How hard is it to factorize sum of two numbers
There is currently no known (asymptotical) more intelligent algorithm (and it is also not expected that there should be one) than if factorizations of $m$ and $n$ were not known (assuming $m$ and $n$ to be coprime). Even the case where $2$ is a prime factor doesn't count, because checking some of the smallest prime numbers can be done without much effort anyway.