I'm going to New York next month and I already have an approved ESTA. Because of the ESTA rules that say I need proof of onward travel, I'm going to book a flight to Brazil or somewhere in South/Central America (Gap Year travels) for just before the 90 days ends. My question is, it will save me about £300+ to get a return flight to the UK going via New York rather than a direct flight from South America, so will I be able to apply for a second ESTA about a month after I have had 90 days in the US? I won't be leaving NY airport and I'll have an onward ticket back to the UK so I can prove I'm not trying to stay.
TL;DR: You're fine.
Long version: You appear to be confusing ESTA with the Visa Waiver Program (VWP), which is common, because this is confusing.
ESTA is a pre-authorization that lets you board flights to the US, nothing more. One ESTA is valid for two years and permits unlimited flights to the US.
The VWP is a program that lets citizens of certain countries enter the US for 90 days or less at a time for business or tourism only. Your VWP period resets to zero as soon as you exit an area composed of the US and "neighboring states" including Canada, Mexico and much of the Caribbean, but not including South America.
So I gather your plan is to stay in the US for less than 90 days, leave the VWP area, then return for transit only on your way to the US. This is fine according to the VWP rules, because your clock is reset by going to South America, and since you will have a ticket from the US to the UK, you will also have a very good reason for returning to the US after that long stay (which could otherwise be considered suspicious). And you do not need to apply for a new ESTA, because your previous one will still be valid.