Is there a way to manually install Google Play store on a factory reset Xperia Play?
See here, you will find all APKs versions of Google Play store.
Go to your phone Settings and see what's your android version.
Download the APK that can work on your phone and install it manually.


Note:


  Remember that since APKMirror is an alternate app source, you must have the setting enabled: "Install Unknown Apps" (or similar), which you will find under security.  – @wbogacz