My information, for placement of the hose, comes from the Service Bulletin #04-06-04-015 (and -015A); which says the NEW hose should end between the battery and the coolant bottle. This AIR pump fix applies to Blazers, Jimmys, Sonomas, Bravada with 2.2L and 4.3L engines from 1999 to 2001 (or, 2003).
How urgent is the repair? I gave a hint, when I quoted the repair manual that the AIR pump is active during cold start-up period (until the engine reaches 160F degrees). I would get the pertinent information and do the prescribed troubleshooting BEFORE changing some parts. Alldata.com is a good source for the troubleshooting charts for P0410. And, it’s free if you go to a public library and go to their subscription web site; or, you could buy a subscription.