What is sticking out from the front passenger side right wheel?

see more .

It looks like a wiring harness and the windshield washer bottle. From the picture it appears the fender inner liner was removed or torn out by ice or snow. You would need to clean the area out and then inspect the tank to see if it has any broken mounts or if it's only the harness. Then determine where the harness goes. It might be the one for the washer pump at the bottom of the tank. I would go to a salvage yard and get all the pieces from another Corolla. Then pull the tire and swap them out. Then install the inner liner. Price wise depends a lot on where you have it done and how much the parts were. Because it's a 2004 used it will likely be easier to find.

Can I just tape the hanging wires as a temporary fix if the windshield washer works fine?

Maybe, clean the wires off and post what the colors are, and we should be able to tell what they are. There are a few that could be there depending on the options. If they are nothing critical and you just need to drive it through the winter, then just stuffing them out of the way could work. However, you really should replace the liner at least, the snow and salt going in there can cause a lot of damage.

I see blue and red. Does this help?

Do you know what is sticking out from the front passenger side wheel? I don't know how much will this cost?

Can you see where the wires go, to a running light? Do all of the lights work, or do you have any warning lights on? You will need to get a new inner fender liner. ASAP. The wiring looks like it needs to be retaped is all. It should not cost more than $100.00 at a shop.

Blue is related to the wipers and washers. If the wire isn't broken, they can be taped up, then install the fender liner and it should be okay.

Thanks so much guys!!

You are welcome, please use 2CarPros anytime we are here to help.