Additional information is always helpful to us in aiding our remote diagnosis. As described, your App Store App is clearly not performing as expected; if the described profile icon (this being a simple stylised “head and shoulders” graphic) is missing, this might suggest a corruption occurred as a result of the update.
What is currently unclear in my mind is whether your problem affects only your iPad - or iPhone and iPad. If the latter, corruption of the update and following remedial steps are unlikely to be appropriate - as it is highly improbable that both devices would suffer the same corruption - and again suggests a local network connectivity issue.
If the problem affects only your iPad, then follow along this path...
If a forced-restart did not clear the problem, there are two ways forward. For both I would recommend that you initially ensure that you have a current iCloud or iTunes Backup of your device.
When you are ready to proceed, you should attempt a full reset of your device settings. This initial reset will not erase any data on your device:
Settings > General > Reset > Reset All Settings
Your iPad will automatically restart. After the reset, all custom settings will need to be set-up again - but for now, concentrate on the problem. Check to see if your problem has been resolved before trying the next step. If all well with App installation and updates from the App Store, you can proceed with manual setup of your device to your liking - and ignore any subsequent steps.
If the previous reset was unsuccessful, we move to the next stage. This step will result in deletion of your local data (of which you should have already created a backup):
Settings > General > Reset > Erase All Content & Settings
Your iPad will restart - and invite you to set-up as new. At the end of the process, you have the option of restoring your data from your backup.
When your iPad has finished its initial setup, I again suggest that you concentrate on verifying a fix of the problem before getting bogged-down in recreating your preferred settings and logging-in to other services. As before, if all appears well with App installation and updates from the App Store, you can proceed with manual setup of your device to your liking - and ignore any subsequent steps.
If still unsuccessful, you’ll need to consider the full system restore process - which will install a fresh copy of iPadOS 13.4 and reset your iPad to its factory-state. Again, at the end of the process, you can optionally restore your local data from your backup:
https://support.apple.com/en-gb/HT201252
I hope this guidance is helpful in resolving the problem.