The part you have boxed is called the air intake resonator. OEM part number is 17230-RAA-A00. This piece is part of the air cleaner setup. It's purpose is to help quiet down the exhaust noise of the engine. As it serves no emissions purposes or engine management functions, you can completely remove this unit with the only affect being a very slight increase in exhaust volume and growl.
As far as repairing the unit goes, I don't know the extent of the damage but even something as basic as a patch using duct tape should be suitable. Please note though that, although it may be the handyman's secret weapon it is not an excuse for a proper repair. If you do decide to go this route, however, do make sure that any patch is one that will not get sucked up into the air system itself. It is before the air filter so any foreign material trying to enter the engine via these damages would be stopped.
Long story short, the prices seem to average about $100 for a new unit at the time of writing - from dealers and OEM online sources. The part is not necessary for proper vehicle operation nor is it, for any reason, legally required to be present. A repair is most definitely possible though repair longevity is questionable.
Air cleaner diagram
Best of luck!