Spray foam insulation for noise reduction can come in two forms; Rigid foam and semi rigid foam.
Rigid foam will insulate against, noise transfer, heat loss, vapour (moisture) and allergens. It is excellent at noise abatement but as any individual who lives in a property that has been built using foam boards for insulation will confirm; it also keeps in noise which can reverberate around a room and make it echo. Please be aware also that No form of Rigid foam will breathe so special construction requirements must be met to use this product.
Semi rigid (or open-cell) foam is breathable so can be used to insulate against Noise, heat loss, and allergens but not vapour (this shouldnt matter if a vapour menbrane is in place). As it's structure is different to that of rigid foam it will soak up noise and not allow a room to echo.
It cannot be pumped into cavity wals, but it can be poured. It will fall to the bottom of the cavity as a liquid and then very slowly expand (appx 4 sq ft in 45mins) to completely fil the cavity and make it airtight.
The other more intrusive & expensive option is to batton the inside, then spray foam between the battons and then lay plasterboard on top. This will mean however that you will have to move any electrics/plumbing etc on the wall.