What is the race order of who has the most wealth/influence currently in the TES world?

It really depends on the context, time and region.

In ancient times, the most advanced civilizations were all elven: the Aldmeris, Ayelids, the Falmer and of Course the Dwemere. Men usually lived in nomadic tribes (the Nedes) and sometimes they were kept as slaves and servants of the local elves, such was the case of Ayelids.

In modern times, I'd say racial prejudice in each province will determine which race is kept in high regards and which is despised. In the cosmopolitan hearth of the empire however, prestige, wealth and power are more important factors.

The races suffering the most segregation outside their provinces are the beast folk and the orcs. The orcs actually don't even have a province that can call their own. Redguards and Nords may concede that Orcs are worthy adversaries in battle but the two races aren't actually on friendly terms. Kajit are considered by the other races as lowlifes and natural born thieves, sometimes even without good reason.

Dunmer are considered the least elven race due to their history. The High Elves consider the Wood Elves their smelly little cousins but still they have them in higher regard than Dunmer and men races. Redguards and Imperials are accomplished sailors and soldiers of fortune, so probably their views of other races are a bit more pragmatical. Argonians generally keep a low profile but they're also strongly sectarians among themselves. Bretons are quite wealthy and known for their constant political infighting which probably contributed to their fame of being untrustworthy and shifty.