I see two problems with the mink photo, poor focus and motion blur.
Looking at the Exif info in the photo, it used "AF Area Mode = single area (0) AF Point (Area Selected) = center (0)",which is exactly what you'd want, yet it focused on the background. I'm wondering if all that happened is that you missed, and the animal wasn't in the AF frame, or it moved.
With a lively subject around vegetation, sometimes it's better to focus on something at a similar distance, and lock the focus before taking the shot. Sometimes only manual focus will get a decent result, which was my main motivation for getting an SLR. MF can be very fiddly on compact cameras. It doesn't have to be, but that's how they design them.
It was taken at 1/60, which is too slow at what it says is equivalent to a 678mm focal length. It had the aperture wide open, so no improvement is possible there. It used ISO 259, so I guess if you use something other than auto mode you could have raised that. I'm not sure how grainy the higher settings are on that camera, but grainy is better than blurry.
It looks like a fairly dark place. As I walk around, I occasionally adjust the ISO up or down so it's ready. Still, if the focus was right, it would have been way better than it is even with the ISO as it was.