How do natural disasters affect politics?
They fundamentally test a government's preparedness and capacity to deal with non-normal situations.
Ideally, governments should be aware of natural disaster risks in their specific region and set policy or regulatory measures in place to deal with them, BEFORE they occur. Progressive and wealthy governments usually do a risk assessment and then act accordingly.
Poorer government may be aware of risks, but have little financial resources to put preparatory measures in place before something bad happens.
Irresponsible or unprepared governments tend to go into react mode once a disaster hits and people usually suffer or die needlessly as the response is often slow or inadequate. In some countries where the government has not acted proactively or done a poor job during a disaster, they may be voted out (in democratic countries at least) or in poorer countries political unrest may follow.