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1. The container in the photo is the engine coolant heat recovery system canister, and the pump adjacent to that canister is the engine coolant heat recovery pump. It runs for a few seconds upon startup and shutdown. Its purpose is to speed up engine warmup from a cold start, to reduce exhaust emissions output.
2. There should be two wires in the connector you circled. According to the wiring diagram, Pin 1 is white/black and connects to body ground. Pin 2 is blue and provides 12V power from the CHRS relay.
3. You can download the electrical wiring diagram at techinfo.toyota.com which is a subscription website.
4. Although that wiring harness connector may have been damaged in the tire blowout, that is not the cause of the AC system not working. If you think it is, you will need to look deeper.

 

That wiring problem will give you a check engine light until it's fixed, but like Patrick says, it won't noticeably affect your drive.

And the check engine light, which presumably is lit now, will give a technician with the proper Toyota diagnostic scanner a complete list of the circuits that need work.

Like Patrick says, to fix any wiring harness problem, the technician will absolutely need access to the manufacturer's wiring diagram and the correct terminals. Good luck with this one.