I think the main reason a country is in poverty is because there are flaws in that country's government system, and that flaw is basically corruption. If corruption can be eliminated from this one country, then poverty can also be easily eliminated from that country.
If corruption is eliminated, the money in each sector will be used properly in that sector. For example, if there is no corruption in the education sector, if the students receive proper education, they will lead the country from within to free it from poverty.
So in my opinion, to eliminate poverty from a country, corruption must first be eliminated.
Corruption is actually the problem of underdeveloped countries swimming in poverty. From mismanagement of public funds to embezzlement of public funds for personal use. This funds embezzled are meant to keep the system functional but due to corruption, few individuals make themselves rich and leave the larger population in poverty. If you check the record of countries with least poverty rate, they also have least cases of corruption.
I buy into your idea that to fight poverty, a country must first fight corruption. It can not be completely eradicated but it can be reduce to a barest minimum which means poverty can be reduced alongside.