As we all know automatic exposure can be very unreliable and often results in blown highlights if left on its own. So what photographers do (at least now that we have digital cameras) is take a photo, see if it looks good, and then adjust the settings if it doesn't.
But why can't the cameras do the very same thing? Take the picture, analyze the resulting exposure, adjust the settings, and then continue taking pictures with proper exposure settings.
Do cameras have a mode where they take a picture, check if highlights are correctly exposed and then adjust the settings?
Kind of. The mode is called 'automatic exposure'.
Your mistake is to consider blown highlights or clipping as 'unreliability' in the light metering, which in most situations is not the case. Except for under the most tricky lightning conditions, any halfway decent camera has a light meter very well capable of realizing already before taking the picture, that the chosen exposure settings will cause clipping, either in the highlights, in the shadows or both and to make a deliberate choice somewhere inbetween to minimize clipping.
If the scene you are trying to photograph has a higher dynamic range than what the sensor is able to capture, the camera must make a best effort choice and guess where to cave in, clip the image and lose details in highlights or shadows. This decision may not be what the photographer finds best, but that is what you have exposure compensation for.
If the camera had just recognized clipped highlights and reduced the exposure to avoid clipping on the high side of the image, the adjustment would have increased clipping on the low side and caused you to lose more shadow details and that may also not be what the photographer wants. The point is, that it does not really matter if the camera makes the decision before taking the picture (as when using automatic exposure) or if the camera tries to improve a botched exposure by adjusting the settings and taking a new picture automatically (as you seem to want). In both cases, the camera must compromise and choose between loss of highlight or shadow details.
Do cameras have a mode where they take a picture, check if highlights are correctly exposed and then adjust the settings?
No.
More precisely, I'm not aware of any commercially produced cameras that do that. What they do instead is to highlight areas that are "blown" in a given shot. You can then decide how much you want to compensate for that and adjust exposure settings or just turn up or down the exposure compensation.
If I'm shooting in Aperture Priority I'd rather not look at the screen after every shot.
There's little need for that. Lighting generally doesn't change that much between shots taken over a short period. If you've used exposure compensation to eliminate blown highlights in one shot, the same setting will probably be appropriate for subsequent shots taken in the same general direction.
DSLRs typically feature several metering modes so that you can choose between accurately exposing one particular part of a scene and balancing exposure across the entire scene. If you're shooting in an automatic mode and you're concerned about blown highlights, consider using "matrix" (Nikon) or "evaluative" (Canon) mode so that the metering system looks at the entire scene rather than just the center point or region.
My cheap DSLR used to constantly over-expose things. Then I bought a much more expensive DSLR, and now it seems to not over-expose things any more. Which isn't surprising, considering one of the main differences on the spec sheet is a much more complicated light mattering system. So to some degree, just buying a camera with better metering helps a lot.
You can also often set the camera to do "exposure bracketing"; that is, the camera takes several shots in rapid succession, each with a slightly different exposure. Hopefully one of these will turn out how you like it.
And then of course, you can always shoot in raw format for the extra dynamic range, and try to fix it all in post.
But is there a camera that takes a shot, analyses it, and decides whether to take another shot? Not that I know of.
I'm not aware of any camera that adjusts automatic or semi-automatic exposure settings for subsequent frames based upon the results of the previous recorded frame. There are many different ways of measuring a scene upon which various cameras base their automatic exposure calculations. Most cameras each have a plethora of metering/exposure options.
But as far as I am aware, all cameras treat each metering event as a discrete one that is not influenced by the previous image recorded.
What you seem to be after is a metering mode that places the highest priority on not blowing the highlights. This is important for many photographers and many lighting situations. But there are also other times when getting the shadows or the midtones properly exposed is more important to the photographer. At those points where the dynamic range of the scene exceeds the dynamic range of the camera, the choice has to be made whether to clip highlights, shadows, or a bit of both.
Canon has long had an option called Highlight Tone Priority for their digital cameras. Nikon has pretty much the same thing called Active D Lighting. There are differences in the way each is implemented, but both start at exposing lower to avoid blown highlights and then pushing the tone curves of the shadows and mids in raw conversion. Other camera makers have offered similar options.
More recently there have been even more sophisticated methods introduced for avoiding overexposed highlights. In their higher level models both Nikon and Canon have introduced newer metering/exposure methods that make avoiding blown highlights the highest priority. Nikon calls it Highlight Weighted Metering (it is only available with D, G, and E lenses - it can't be used with other lenses, even if they are "chipped"). Canon still calls it HTP, but the way it is implemented is far more sophisticated than early versions, and the versions currently used in the lower models, of HTP.
Beyond that, the introduction of RGB or RGB+IR light meters with hundreds of thousands of pixels divided into hundreds of metering zones has enabled very sophisticated "library" based metering where a metered scene is compared to a stored library of typical lighting scenarios. This has further refined the camera's ability to more often correctly guess when the highlights are more important and when the shadows are more important. To some degree such light meters are miniature imaging sensors. Meter readings, though not recorded to the camera's memory card, are effectively low resolution images upon which the camera bases exposure decisions similar to what mirrorless cameras (or DSLRs when shooting in Live View) do using the camera's main imaging sensor.
In the end, there's no such thing as correct or perfect exposure. There's only the image the photographer wishes to make. What may be perfect for one may be overexposed for another and too dark for yet another. Most photographers who are critically concerned about getting exposure "just so" will eventually learn to use Manual exposure mode so they can have complete control over all exposure parameters.
