When creating Master Password, one of the primary goals was to create a password app that does not send private information over the Internet. As such, your Mac does not send your newly added sites to your phone over the Internet. Anything you create in Master Password stays on your phone (or computer) and doesn't leave it.
That said, the beauty of Master Password is that if you add, for example, twitter.com on your Mac first, and are later looking for its password on your iPhone, all you need to do is look it up on your iPhone, create it, and the password will be the same as the one on your Mac, even without sending this private information over the Internet.
To make a long answer short: your phone's new site will not automatically show up on your mac, but just add it to your mac the first time you need it and the right password will automatically show up.
What I do about transfering all my credentials to another device is simple:
In my case, my credentials are created first on my iPhone.To transfert all the stuff, in MasterPassword app I go to "Export, Export Sites, Safe Export" and send the file to the Files app.
That file should show up on your desktop computer in iCoud Drive (if iCloud Sync in enable of course).
Then go to your Mac, launch your MasterPassword program and import that file.
Every site with their password, password type, counter value will be there.
The only thing that is not synchronize is the user name unfortunately, it reverts to auto genrated 9-letter name. May be the developer could shade some light about this omission.