The President should solve issues that are not popular or politically correct because the job of the president is to lead the nation. The President was elected by the people to fulfill a duty; therefore the President should work to solve whichever problems need solving most, regardless of if they are politically correct, instead of only focusing on re-election and working on problems that won't upset voters.
The president should not worry about political correctness. So solving these matters is a non-issue. He should focus on the financial health of his country rather than trivial matters of speech and writing. If an issue is popular it will be addressed anyway, but a problem that is not popular might deserve attention as well.
Of course the President should solve such problems. Why else did we vote for him? Or what is the point of having President? To prance around and hold empty speeches and occasionally clash with the Congress over some political nonsense? No, he should be fully involved and devoted to such problems and use his influence and the trust the people have stored in him to make such cuts and steer the State and nation to safe ground.
Yes, the president should solve issues that are not of the highest concern to all of the citizens. A president cannot solve world hunger and the economic problems in a few days or even in a few years. There are a lot of people that need to approve ideas and processes that bills need to go though. In the mean time, why not take care of issues that CAN be solved?