What is important to have about a car when doing circuit racing?

May be...

If you are looking for what makes a great racecar... in short: everything. The main limiting factors will be 1) budget, 2) rules. You mention the Miata... the Miata ia a great track car for many because it is lightweight, well balanced, quite inexpensive, large numbers of the car were made which means replacement parts and bodywork are easy to get from salvage yards... and, there are many classes where a Miata can fit in and be competitive (even aside from the Spec-Miata class).

As a generalization, power is important, but if you take a car and look at what will make it do a faster lap, you will generally find it takes quite a bit more power to make large gains as where smaller improvements in the suspension or setup can make much bigger gains. Tires will make the biggest plus, and many races are decided by who is running which tire and what compound. Next comes having a good setup (appropriate alignment, corner balancing, etc). Connected directly to that are adjustable suspension components (Anti-Roll Bars, Shocks, Springs, etc).

Lighter is also always better... even getting well under the minimum race weight since when adding ballast weight, you can choose just where you want it. Small things like a good steering wheel or race shell can make a big difference too by lessening driver fatigue. After all of that, then you get to power! Power can make a big difference when everything else is sorted out... and then, small gains of a couple hp can be just enough to get the job done (since in most races, the rules are such that the cars are quite evenly matched).

I know they even race mx-5 meita's so what is importent because power is only good for 1/4th racing in streight line.