If I cover two-magnet setup like the one depicted with horseshoe made of mumetal, will the returning paths of magnetic field be mostly enclosed in the horseshoe or will the returning paths in 3rd dimension (not depicted) stay unaffected - with the same strength? How will this cover made of mumetal affect the primary paths of the inner part of the magnetic field? Will the flux get stronger, weaker or stay the same inside - between the magnets?
The thicker the mu-metal the more the flux will return inside your horse-shoe construction. The inner field will increase for the following reason: you may think of this in terms of magnetic charge. From left to right you have (-)-(+)--(-)-(+) at the surfaces of your magnet. The outer surfaces, hence, produce an opposite field. The mu-metal would sort of spread this surface charge into 3D, therefore, reducing the according field.