Usually what you terms as grainyness is to with the ISO setting. This basically the sensitivity setting of the camera, and it the higher the value for the worse the grainyness is. Here is an in depth explanation: "https://photographylife.com/what-is-iso-in-photography" (sorry cant post links yet). The ISO is set automatically by the camera app.
The reason for this is usually poor lighting, and under artificial lighting it can also get bad. What I suggest you do if try going outside on a reasonably sunny day and taking a video and see if the grainyness is still there, where you have good lighting. Alternatively if you have a camera app which allows manual camera settings you could play with the ISO setting and then you would see what I mean.
Personally I don't think it looks that bad, if you look at the floor at your puppy they don't look too grainy for me. However I am not expert and I don't own a MXPE so I could be wrong about this. What I would say is have a go maybe outside where the lighting is better and more natural.
Also P.S. the grainyness can, I think don't quote me on this, come form the compression of the video and will depend on the type of file which is outputted by the camera (e.g MP4, AVHCH etc..) although am not sure if you can control this.