The "light spots" are just overexposed portions of the image, mainly the lighting and reflection on leaves facing the incoming direct sunlight. "Dappled shade" is notoriously difficult to expose for when the image is taken during bright sunlight (as opposed to morning or evening). You are fighting exposure for small areas of intense light falling on your subject and the shaded areas of the rest of your subject. The range in exposure between these two extremes defines your working dynamic range (that is, areas lighter than the brightest areas you want to preserve are allowed to blow out to pure white and areas darker than your darker detailed shadows can clip to black).
As proposed earlier in the thread, you can attempt to bring the two extremes closer together by using fill flash - this will bring the shadows closer to the highlights and compress the dynamic range you are trying acquire in a single shot. Otherwise, you will need to make a decision about where your exposure should fall to make the image you want.
Usually, the most objectionable image element is overexposure on the important skin tones of your subject - for example, the side of a face that blows out to pure white with no hope of recovery. You may have to shoot with this portion of the face placed in the headroom of the highlights for your camera and let the rest of the scene fall where it will. You can also try to reposition your subject so that they do not receive the difficult lighting and the exposure on them is more uniform (for example, all shade).
This is all predicated upon the assumption that you are shooting raw. If you have not, take the time to experiment with your particular camera and software combination to see just how far you can extend your (over)exposure before you completely lose data to clipping. Some raw converters have pretty sophisticated highlight reconstruction algorithms, but even those may look unnatural if pushed too far.
If the area in the image is important but really small (a small sliver of a face lit by direct sunlight) you may want to let it blow out in order to expose for a larger portion of the subject. Get to know how much dynamic range your camera can accommodate in post, and how far you can push shadows without revealing objectionable levels of noise and other vagaries of underexposure.
Exposure is a balancing act. There may be reasons why you would want to capture the entire dynamic range (like real estate interior photography) - in your case, spontaneous portrait shooting would really require careful exposure in a single capture, with software that permits you to squeeze the most out of your image data. At some point, you will have to let highlights blow and/or shadows block up.
Also, highlight reconstruction is sensitive to white balance, so make sure you understand how these two factors interact when working with the extreme highlight range of data in your images. All else being equal, I would think skin tones are most important, so concentrate on how these tones get rendered in your experiments, perhaps (balancing the green cast that often falls on subjects under tree leaf canopies, for example).