I have a 2005 TB and also had a 4010 code. My story is in another thread on this forum here: Secondary Air Injection Pump
If there was no water in the line down by the pump then the check valve on the engine exhaust manifold is either working correctly or it is stuck closed. My check valve was stuck open and the hose had plenty of water.
The other problem could be the pump itself. Since you are getting intermittent codes that clear themselves your pump may either be working intermittently or it may not be putting enough pressure into the hose.
While you're at it make sure both hoses are connected securely and that there are no defects like kinks, cracks or breaks. An open hose connection or break in the line from the pump to the check valve would be the same as if the pump was not providing full pressure.
I pulled the pump and put power to it and discovered it was not running at full speed. It also had some "gunk" in it.
I ended up replacing the check valve (since I knew it was bad) first. When the code came back I replaced the pump. The code cleared after the next cold engine start and has not returned.
Replacing either the check valve or pump is easy. You will need to raise the vehicle some to work on the pump. BTW I bought a new pump via the AC-Delco website and got a price lower than my dealer and local parts store.