hello and welcome to just answer.
There is an extremely lengthy diagnostic chart that I can provide for you, however unless you are familiar with todays electronics and poses a TECH-2 scan tool it would not be much help for you.
There is also a bulletin that I will provide, it states that an oxygen sensor could cause this problem. I would speculate that if the air pump operates correctly replace the o2 sensor and retest the system.
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#PIP3530: Repeat SES Light Due to a P0410 DTC - Possible O2 Sensor - keywords AIR CEL injection MIL oxygen pump secondary - (Aug 24, 2005)
|Subject:||Repeat SES Light due to a P0410 DTC - Possible O2 Sensor|
|Models:||2004-2005 Buick Rainier |
| ||2004-2005 Chevrolet TrailBlazer |
| ||with 4.2L Engine (VIN S - RPO LL8) |
The following diagnosis might be helpful if the vehicle exhibits the symptom(s) described in this PI.
Currently, the SI diagnostics do not mention the position 1 O2 sensor as a potential cause of a P0410 DTC.
If the SI diagnostics do not isolate the cause of a P0410 DTC, there is no sign of dirt or water in the AIR pump or relay, and proper AIR check valve operation is noted, replace the position 1 O2 sensor.
Please follow this diagnostic or repair process thoroughly and complete each step. If the condition exhibited is resolved without completing every step, the remaining steps do not need to be performed.