ChuggingAlong

When I accelerate above 80km or 100km (different each time) the Check Engine Light goes on.

When I run the scanner tool it comes back with the following code:
P0420
Catalyst System Efficiency Below Threshold (Bank 1)

Is there any way the most basic non mechanic person can check to see what may or may not be faulty?

Background:
In 2009 my drive light flashed one morning and the speedometer went wacky (not reading the correct speed but jumping all over) and to accelerate took a few seconds before it would slowly start moving.

I took it to a Honda dealership and they said the wires had been eaten through by mice. After replacing the wiring harness etc (over $3000 worth of "repairs") they said it was fixed. The next morning it did the exact same thing. The car was in the shop for a week and once "fixed" they didn't bill me nor tell me what they did to get it working. We didn't part on good terms.

The P0420 kept coming on and I learned to live with it because I wouldn't take it back to them. Everything else was fine until this past winter (northern Ontario/Canada) when it was cold in the morning the drive light would flash until 10 - 15 minutes of driving (same problems as previously) until I drove above 80-100km. At this point I would stop the car and reset the codes and it would drive fine.

I don't have the codes (numerous codes) from the winter but consistently the P0420 comes on.

Normally I'd take it to a mechanic to get it fixed etc., I have almost no car experience, but finding myself out of work and with zero money that's entirely out of the question.

Any tips/help would be greatly appreciated. If any more information is needed I can supply it. Thanks in advance for at least reading this even if no responses are given. I do appreciate it.