Can savings and poverty coexist? Many governments try to decrease savings in the economy so that recessions do not occur. Does it make sense for savings and poverty to coexist though at significant scales?

Let's say a population was both in widespread poverty and they had savings. If they had unmet needs, they would spend their money on those needs. This would occur until either savings where expended or the needs where met.

If savings where bad and caused poverty, it would eliminate itself it seems. So either it is bad, but doesn't need to be discouraged since it eliminates itself, or is good and doesn't need to be discouraged.

I guess the problem could be if savings where uneven, but then


That doesn't mean savings are universally bad, just savings of the non-poor.
If there is a recession, people will have less money to buy things, and so things will need to be sold for less. This will encourage people with savings to spend.
First we must address the nature of savings. Savings are not bad or good per se. It depends on the context. It is true that in the short run, increased savings will lower consumption, thereby lowering GDP. However savings also increase investment, which in the long run increases economic growth. Therefore, the optimal savings rate is found by solving the trade-off between the short and long-term. 

If you are interested only in the short-term (GDP right now and a few periods more) then you would see savings as bad. If you are interested only in the long-term you would see savings as good. If you are interested in both, then it is ambiguous.

The optimal savings rate is certainly not 0 for a standard economy. One way the optimal savings rate is characterized is through the so-called "golden rule" in a simple Solow growth model. You might want to look this up.

Empirically, we typically observe that most countries save too little, i.e. are below the golden rule savings rate. If you are below the optimal rate, whatever that may be, then more savings are good. If you are above it, then more savings are bad. Hence, it is not straightforward to make a case about the desirability of savings.

Your second question was about poverty and savings. Yes, they can and do coexist empirically. In fact, many economists are puzzled why savings rates are so low in poor countries, since more savings increase growth over the long-term. 

To see why savings and poverty can coexist, you must think about the reason we have savings. Individuals save more today, so that they can consume more tomorrow. Hence, you cannot really detach the two completely and say that savings are bad. Imagine a poor country that is primarily agricultural, as is often the case. Farmers there make money a few times a year, maybe once or twice, when the crops are sold. However, they must live off this money for the whole year. The only way to make sure you can survive the winter is to save money from crop sales. Furthermore, the poor also need to make big purchases in the future and the only way to do so is to save. E.g. they are poor, but must save up for a wedding, etc. Or they need to buy fertilizer for the next season and the only way to do so is to save enough money. All these things are possible as long as death from starvation is not an imminent threat, which for the majority of poor people today it is not.

Savings do not exclusively occur when all other current consumption needs are totally met, as it seems to be implied in your question. In fact, they can never truly be met, as ressources are scarce. Savings are instead determined by the optimal decision of how much to consume today vs. tomorrow, or how many of my needs to leave unfulfilled today vs. how many unfulfilled tomorrow. Hence the fact that poor people cannot fulfill all their needs today does not imply that they should not save, since they also think about the future and how to fulfill those needs tomorrow. Think of surviving on an island with limited food. You would likely consume a bit everyday and be hungry in order to survive, than eat everything in one day, be full and then die in a few days.