Hello, sorry to resurrect an old post. P0410 is secondary air pump error code. There are three components that contribute to the error. 1, the little blue valve under the engine cover in front of the secondary air pump. Check and make sure the little ball inside is freely moving. 2. The RCM or relay control module, you did a good job by resoldering it, but that's not enough, if the air pump has gone bad you'll need to clean up the number 5 relay (or number one depending on how you view the universe) see video below. 3. The pump itself. As seen in the video my relay was shorted engaged so the pump was always running which contributed to a shortened life. Unfortunately, if your scenario played out like mine, you will likely need a new secondary air pump.