For a 1060 GTX which about 30% faster than the RX 470, 4590 would only bottleneck it worst case scenarios. If you are getting FPS drops, check your CPU usage to see if it's hitting 100% at the time of the drop. HWInfo is good for this and it produces graphs. If you are hitting 100% usage or generally are playing over 90% usage then your processor would be bottle-necking.
Check your background CPU usage and see if you have applications eating up CPU resources.
A processor like a 4790k would be able to handle increased workloads better and which would in turn lead to smoother game-play.