Hi!
Thank you for the details, it helps us a lot.
Sorry if I was not clear, what I meant is to disable the Ajax Search Pro plugin (not the other one), to see if the issues persist. If yes, then it means that the plugin is not related to the problems. I no longer see however the notices on the front page, but I still tested the other cart issue.
I have disabled the ajax search pro plugin, and the Cart icon is still containing the wrong URL, causing the problem. This means that the issue is not related to the search plugin, since it persists even if it is completely disabled, see the screenshot: https://i.imgur.com/gSZEm8b.jpg
You can test this by disabling, or even uninstalling the search plugin, the issue will still persist.
The Cart URL error message is thrown by WooCommerce itself, saying that the function which prints the cart URL in the Theme is the outdated version. Usually ‘Notice’ type of errors are not displayed in WordPress unless the debug mode is enabled. Perhaps on your staging environment you have enabled the debug mode in the wp-config.php file. These issues can be present in the live environment as well, but if the debug mode is disabled, these are not visible.