Corolla,Can the power steering return hose problems cause the Check Engine Light to come on?

My 2004 Corolla apparently needs a power steering return hose replacement according to the dealer. I have had check engine light turn on randomly during the winter (Illinois) before and I was told by several repair shops that the vibration of the engine when it starts up on cold days is setting off the check engine light and that it really doesn't need any repairs other than resetting the code. The check engine light came on several weeks ago and I was wondering if it could have something to do with the power steering or if it could be just some other random issue.

My car has 119000 m