You can set up a direct login link from YourSites to the backend of the client site.
In site entry in YourSites add a user id or user name that you want to auto login with. Note that if you need two factor authentication to login and configure the client site plugin to require it then you would need to provide that when you click the direct login link
This can be disabled on the client site or additional security constraints implemented. Edit the client site YourSites plugin and make sure you haven't disabled direct login. While you are there you could take a look at how you can restrict access to this plugin based on referrer, IP address etc.
Note that this direct login can handle backend access controls using htaccess username/passwords and should work with plugins that set and access key for backend access to your site.
We love every single one of our users, without you YourSites simply couldn't happen! So we would love a review at the Joomla! JED so we can let others know about us too, please take a minute to write a review:
If you feel you have something negative to say, we would implore you to speak to us first, as we really really don't want anyone to be unhappy!