Do axon grows after cutting/damaging of some of its part?
Well look at this picture from Junqueira histology book:

In an injured or cut peripheral nerve, segments of axons distal to the injury lose their support from the cell bodies and degenerate completely. The proximal segments can
regenerate from their cut ends after a delay. The main changes that take place in an injured nerve fiber are shown in the picture above.
In the nerve segment distal to the injury the axon and myelin, but not the connective tissue, degenerate completely and are removed by macrophages. While these
regressive changes take place, Schwann cells proliferate within the connective tissue sleeve, giving rise to rows of cells that serve as guides for the sprouting axons
formed during the reparative phase.
The axon grows at the rate of 0.5–3 mm/day