Probably not much. A common darkroom rule is: expose for shadows, develop for highlights. Push processing increases development, which mainly boosts highlight density and overall contrast; it does not restore shadow detail that was never recorded because of underexposure.

With HP5 underexposed by around 4 stops, pushing may produce something visible, but results are usually poor if you want a normal-looking negative. Expect very thin shadows, blocked-up contrast, and pronounced grain. In other words, development can make the negatives easier to print or scan, but it cannot truly recover missing detail in dark areas.

If the images matter, it may still be worth trying, because some frames can sometimes be salvaged during scanning with careful adjustment. But for a severely underexposed roll, keep expectations low: usable in a rough, high-contrast way is possible; clean, normal results are unlikely.