What you may try is reverting the Google Play Store app to the default version. Maybe the update was downloaded improperly (e.g. your device got cut off from the internet for a moment without you being aware of that). You can do it, by following the steps mentioned below:
Go to Settings and tap on Application Manager/Apps;
Tap on Google Play Store, and select Uninstall updates;
Tap on OK to confirm;
Open the Google Play Store app;
After the updates have been uninstalled, you may try to install them again (optional).
Also, what you say suggests that there are files of the app somewhere, I suggest you open the file manager and clear them, and then try installing the app anew.
One more thing you can try is wiping the cache partition via the recovery mode (very likely to work):
1. Power off the device. If you can't do that using the power button, just remove the battery and insert it after 10 seconds.
2. Press and hold the Volume (+) up and the Power (Sleep/Wake) buttons. Once your device turns on, let go of the buttons.
* If you use a Samsung device, press the Power (Sleep/Wake), Volume (+) up and Home button.
3. Press the Volume up or Volume down key to navigate through the options. Highlight the Wipe cache partition then press the Power (Sleep/Wake) key to make the selection.
Source:
https://solutionbay.com/solutions/how-to-fix-error-code-501-on-google-play-store