While a better camera (read DSLR) will likely give you a better photo, I am guessing you'd like a better shot with your phone. Getting rid of all the grain will be difficult because the ISO is likely getting pretty high, but balancing the light in the photo may help.
In the composition you showed, you can see there is a fair amount of ambient light. I would suggest not having too much light in the foreground, which will make it easier for the camera to meter the shot. Compose your shot to have the moon be the brightest light in the photo. Use the the finger focus point to move the circle around until you find a spot that gives good balance to your photo. Too close to the moon (light source) will result in a very dark picture everywhere else, to far and the moon will be "blown out" (too bright). Some trees or dark buildings in the foreground may or may not add interest to you photo as a silhouette -- there is no cost to playing.
An example of an exception to too much light in the foreground would e a cityscape. Those provide lighting balance problems that are normally dealt with by merging photos (HDR). I have not played with the Note 4 cam enough to see how the build in HDR handles such shots.