CEL code P0410

I looked up this code and it means something about some water that isn’t draining from a hose properly or something. What exactly does it mean and about how much would it cost to get fixed at a repair shop?

The official definition for this code:

P0410 - Secondary Air Injection System Malfunction.

It could be a failure of a moisture drain, but that guessing until it is diagnosed. The secondary air injection system consists of an air pump, a valve box, and hoses to inject air into the exhaust stream. A system malfunction code means something in the system is not operating correctly. This system is necessary only for emissions, and if you don’t have emissions checks in your area, this is not an immediate repair item. The cost to fix this is highly dependent on what is not working properly, and no one can give you an estimated cost without determining the cause of the fault. Careful of some cost estimates, because they could price out a complete overhaul of the entire secondary air injection system, which would be complete overkill in most situations and probably expensive.

It helps, some people, to understand what the AIR pump does. It is an electric pump, mounted under the radiator. I’m told that it only runs during engine warm up. It pumps air into the exhaust manifold to aid burning of the rich air/fuel mixture incompletely burned in the cylinders.
There are Technical Service Bulletins (TSB) about the problem of an air pump which has swallowed water, rusted internally, and failed. The TSBs don’t contain a thorough troubleshooting section…just, “Change it thusly”.
There are better troubleshooting charts at alldata.com. There might be solenoids actuating the check valves (I haven’t recently re-read the TSBs). One would want to check those.
Hopefully, you can find a mechanic who will troubleshoot the problem, instead of the knee-jerk reaction of, “Let’s change the AIR pump”!