I have a P0410 code?

A leak like this.

Yes, the code is for the secondary air injection malfunction and that hose goes to the system diverter valve. I would replace that vacuum line and then clear the codes. Let me know if that fixes it.

Well, due to the mechanics not working through the weekend, I temporarily taped it up with electrical tape. The check engine light went away on its own before that. While I was researching how the secondary air injection works, I figured out that it only activates when the engine is cold. It helps warm up the catalytic converter in the first 40 seconds of the engine running for emission regulations. This morning, when I turned on my car, no check engine light appeared, so the electrical tape seems to be doing its job for now. Of course, I will replace the hose on Monday. I have a follow up question. Is it possible that there were no check engine lights this summer because it was warm, and there was no need for the secondary air injection to activate?

Yes, that is correct. I would replace the hose and then clear the codes, and the problem should be fixed.