Mobile racing games lag on MacBook Air

Also, do such games generally perform better through native Mac App Store versions compared to browser-based or emulated ones?

I would think so. Browsers and emulators are not "light-duty" apps, so you add the browser's or emulator's overhead to that of the game and OS.

It could be internal heat. Macs have thermal sensors inside and can clock back the processor if the temps get too high due to workload. And remember, M-series Macbook Airs have no cooling fans, relying on convection cooling instead. Some here have issues with Macbook Air excessive heating but others like you have reported it.

If you have added any anti-virus or so-called "cleaning" apps, check your foot for a wound. Both can greatly increase the workload without retuning any benefit. The macOS protects itself quite well.

Those generalities are about all we offer here without more data. If you want a data-driven evaluation in this setting where we can neither see nor touch your computer, please post an EtreCheck report. We can quickly and within the limitations of these forums help you determine what issues are at play without our playing a protracted game of "20 Questions" with you that could go on for days. 

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The free version will do nicely for this purpose, although the app is worthy of our financial support.

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If you’re running the Hill Climb Racing APK, that’s almost certainly why it lags. macOS can’t run APKs natively, so the emulator adds extra load and causes stutter, even on an M2 Air.

A native Mac App Store version (if available) will always run smoother than any emulated or modded APK. The fanless Air can also throttle when temps rise, which doesn’t help performance.

If you want to be sure nothing else is causing slowdowns, posting an EtreCheck report is still the quickest way for people here to check your system for any hidden issues.

Note that Apple Silicon Macs can run iPhone and iPad apps – if the developers of those apps allow it. Although the Mac platform is different from the iPhone/iPad platforms, there is an extreme amount of common technology under the hood, with regards both to the processors, and to the operating systems.

If there is an iPad version of an Android game that you want to play, and the developer permits that iPad version to downloaded to Apple Silicon Macs from the App Store, the iPad version might run better than the Android one.

Here are some iPad racing games that you can run on an Apple Silicon Mac (like your M2 MacBook Air). The following is a screenshot of part of the App Store window.

Note that many of these have the fine print "Not verified for macOS" – which means that the developer has not tested whether they work well on macOS, or maybe even whether they work at all. The ones which just say that they are "Designed for iPad." without the "Not verified" caveat might be a better bet.

Note that when both Mac apps and iPad apps are available for the same type of game, the games designed first and foremost for the Mac might provide a better experience than the iPad ones.