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Hi there!
I'm so sorry for the confusion on this! The API is actually separate to Moz Pro and not included with the free trial. You can see more about our available API subscription levels here: https://moz.com/products/api/pricing
If you'd like to know more about the API or have any questions about the subscription, please email us at help@moz.com
I'm afraid that I don't know of a way to change these other than finding them all and manually amending them. We do let you know the URLs of the referring pages so you know where to look - this is easier to see if you download the CSV file for these 404 errors from your crawl.
I think the Broken Link Checker Wordpress plugin can also offer you a list of these broken links that you can click through directly to amend them but I don't have any personal experience with this.
Sorry I can't be more help!
Hi there!
It looks like these 404 errors are coming from improperly formatted links in your html. I've attached an image of these in the example that you gave. As these links do not contain the http:// to tell the browser that they are external to the current site, the URLs are appended to the URL of the current page.
You'll want to find and amend these links as the 404 is also being seen by users if they try to follow them.
Sorry about the delay in getting back to you! In general, we can't give you a timeframe for the fix aside from "as fast as humanly possible". With an outage like this, the engineering team works pretty tirelessly to get things back up and running but there can be multiple possible causes and we'd rather not break anything else while we solve the issue.
I'm glad the Health Page helped! 
We will keep updating it with the latest info. We are still monitoring this but things are looking a lot better after the fix we deployed at 13:51 PDT yesterday.
Hi there!
I'm so sorry that you're running into this error message!
Unfortunately, we are currently having some problems with Open Site Explorer and, although our engineering team is digging into it, it's taking a little while to fix.
We will be posting any updates on the Moz Health Page so I'd recommend signing up there to be updated when the issue is resolved.