It is impossible to say what is the cause of the blurry photos without seeing examples, but some of the causes:
Not holding the camera steady- if you zoom a lot, the camera needs to be especially steady. A tripod might be the only answer
Too low a shutter speed- Too low a speed magnifies hand movement.
Subject moving- Too low a shutter speed won't "freeze" the subject.
I do not quite understand your second question about multiple photos, but it sounds like you are not using the cameras burst mode, but instead using its image stacking features, which take multiple shots and combine them into one photo. The image stacking modes are for either giving better dynamic range (allowing better photos of scenes with both bright and dark areas) or for lower noise (less fuzzy pictures in low light).
I recently purchased a Nikon P500 and noticed that a lot of the pictures come out blurry. This often happens on the Green camera selection. Based on my explanation, although I love taking pictures, you can see that I'm not a camera buff.
I had purchased a previous version of the Nikon camera and had a similar problem so I'm left wondering if the blurry pictures problem is part of the Nikon camera family.
The other issue I have is that when I try to take multiple pictures, using that feature, it takes about 20 at once and then there is a delay and only one picture is actually left behind. I don't understand that.
I used to have a Sony DSC and that camera was beautiful and I never faced these issues. Unfortunately it was stolen and is no longer made....although not an old version....

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