Hello GM has had lots of issues with the secondary air injection pumps if there is water inside the air pump then it needs replaced. There is a service bulletin on p0410 but when they get any type of moisture it damages them. I myself have tried to lube and reinstall the pump but due to lack of pressure output it would set check engine light. I'll post the bulletin below. I would also inspect the air injection relays for corrosion or water intrusion.
2001 Chevrolet S10 Pickup - 2WD | Blazer, Bravada, Envoy, Jimmy, S10 Pickup, Sonoma (VIN S/T) Service Manual | Document ID:(NNN) NNN-NNNN
#04-06-04-015A: LN2, L43 or L35 Engine SES/MIL Light Illuminated, AIR Pump Ingesting Water and DTC P0410 Set (Replace AIR Pump and Inlet Hose Assembly) - (Nov 28, 2006)
This bulletin is being revised to remove a part number and change the two AIR pump part numbers in the Parts Information. Please discard Corporate Bulletin Number 04-06-04-015 (Section 06 -- Engine/Propulsion System).
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.
P0410 Set (Replace AIR Pump and Inlet Hose Assembly) - (Nov 28, 2006)
Water may have collected in the AIR pump or the vent solenoid and frozen or corroded the pump.
Replace the AIR pump and install a new inlet hose and solenoid tube assembly using the following procedure:
Tighten Tighten the bolts to 17 N·m (12.5 lb ft).
Pump Kit, Secondary Air Injection
Here's the location of the fuse and relay block from what the picture is showing if your standing in front of the vehicle it's on the left hand front corner. Not sure if your S-10 is a base model or a uplevel so i'll post both illustrations. From what they are showing it's a 10amp fuse in the underhood fuse block.