I have a service engine soon light on and have hooked up an OBD11 reader to it and got code P0410 pertaining to a secondary air injection problem. hoses are fine, pump is new.

Secondary air injection system P0410
First check the fuses and relays for the secondary air
Pump, if blown, replace them...otherwise, replace the pump..

Possible causes
- Faulty Vacuum hoses damaged
- Faulty Air Pump inoperative
- Air bypass solenoid harness is open or shorted
- Air bypass solenoid circuit poor electrical connection
- Air bypass solenoid leaking/blocked or stuck open/closed
- Air diverter valve leaking/blocked or stuck open/closed
What does this mean? When is the code detected?
The secondary air injection system does not detect the presence of air in the exhaust when introduced by the secondary air injection system.
Possible symptoms
- Engine Light ON (or Service Engine Soon Warning Light)
- Engine hesitation
P0410 Description
An air pump is used on the vehicle to lower tail pipe emissions on start-up. The Powertrain Control Module (PCM) grounds the Air pump relay control circuit which energizes the Air pump. The PCM also grounds the Air combination valve vacuum control solenoid circuit, which energizes the Air vacuum control solenoid. Vacuum is then applied to both Air combination valve diaphragms which opens the shut off valves. The PCM enables both circuits simultaneously when Air system operations id desired. When the AIR system is active, then Air pump forces fresh air into the exhaust stream in order to accelerate catalyst operation. The AIR combination valves replace the conventional check valves. When the AIR system is inactive the shut off valves prevent air flow in either direction.

Read more: http://engine-codes.com/p0410.html#ixzz31FMsMMLJ

See this link for a common problem...
Oldsmobile Aurora P0410

Diagnostic Test Code (DTC) P0410 indicates that your secondary air injection system is not functioning properly.
Not all 2001 Chevrolet Bazers are equipped with secondary air injection. The secondary air injection system was added to vehicles that were to be sold in certain areas where the emissions laws required it, like the State Of California.

The Powertrain Control Module (PCM) monitors the system by commanding the secondary air blower motor to run and the air injection solenoid to open. This causes additional oxygen to enter the exhaust system. The PCM then monitors the signal from the oxygen sensors to see if the commanded air has in fact, entered the exhaust system. Code P0410 sets when the PCM has commanded the blower motor on and the solenoid open and there is little or no change in the oxygen sensor signal.

GM has had some problems with these systems and there has been a Technical Service Bulletin (TSB) issued that addresses some of the known concers. Below is a brief description of the TSB and the issues involved.

Bulletin No.: 04-06-04-015A
Date: November 28, 2006
TECHNICAL
Subject:
LN2, L43 or L35 Engine SES/MIL Light Illuminated, AIR Pump Ingesting Water and DTC P0410 Set (Replace AIR Pump and Inlet Hose Assembly)
Models:
1999-2003 Chevrolet Blazer, S10
1999-2001 GMC Jimmy
1999-2003 GMC Sonoma
1999-2001 Oldsmobile Bravada
with Vortec 2.2L I4 or 4.3L V6 Engine (VINs 4, 5, W - RPOs LN2, L43, L35)
Condition
Some customers may comment on the SES/MIL light being illuminated. Upon investigation, the technician may find a DTC P0410 indicating that there is a concern in the secondary air injection system.
Cause
Water may have collected in the AIR pump or the vent solenoid and frozen or corroded the pump.
Correction
Replace the AIR pump and install a new inlet hose and solenoid tube assembly...

The bulletin goes on and gives detailed information on how to perform the repair and the part numbers of the replacement parts needed to do so. Your local dealer can provide the parts for you and you can probably get them to print out a complete copy of the bulletin for you. It is not really hard to do if you want to do this yourself.

CEL light/code P0410, secondary air system fault.
DTC P0410 - Secondary Air Injection System

Suspect -
Relay/fuse (box B) loc 3 ,15amps
Pump $434(new) / $243 (Remanufactured) / $95 (used,Ebay)
solenoid $73
check valve $18
http://www.autopartswarehouse.com

A new pump, or simulator!

What signals the ECU, if the seconary Air Injection system isn't working?
H20 sensors, if running rich > Xtime throw code!
The pump only runs when the engine is cold, so the air & rich fuel mix is burnt in the exhaust manifold, so there's less Emission.

P0410 - Secondary air injection system.

https://www.obd-codes.com/p0410 Look on youtube .
P0410 Secondary Air injection System Troubleshooting

P0128 - Coolant Thermostat (Coolant Temperature Below Thermostat Regulating Temperature)

P0410 - Secondary Air Injection System Malfunction
(has to do with exhaust system)

Note: There is a technical service bulletin (TSB) regarding code P0410 for a number of GM models. The TSB calls for the replacement of the AIR pump and the installation of a new secondary injection hose.

most ecu controlled vehicles have a 'limp home'mode if 2 or more fault codes are stored,it is designed to get your attention before damage is caused and to get you to the nearest w.shop to fix,i would have codes cleared and retest vehicle,it is the cheapest and 1st option for any shop,some late models even have excessive mileage past service interval in the 'limp' program

I am very familiar with this code, there is a GM service bulletin about this problem, in short it says to replace the secondary air injection electric pump assmebly