Is fighting corruption the only way to eliminate poverty?
If eliminating poverty were easy, it would no loger exist. Although the world has made great strides in the Millennium Development Goal of eliminating poverty, we still have a long way to go.
For your specific question, there is no one solution to the problem of poverty, as poverty has many causes, which vary depending on region and socioeconomic circumstances.
In some countries, anti-poverty efforts are hampered by corruption, and in those countries fighting corruption would contribute to reducing poverty. Corruption, however, is not the only reason for poverty.
First, a history of racism and colonialism contributes to poverty. In many countries, historically oppressed peoples or classes (such as African-Americans in the United States, dalits in India, the First Nations in Canada, or the Tamils in Sri Lanka) have difficult escaping a cycle of poverty and discrimination.
Next, disease can keep people in poverty, again forming a vicious cycle. When people are ill, they cannot work, but inability to work leads to inability to afford medication.
Other issues have to do with land ownership, environmental degradation, infrastructure and access to education. Successful efforts to reduce poverty usually take a multifaceted approach, addressing the various different causes of poverty rather than just one facet.