OK. I took everyone’s advice, this is what I did:-
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I cleared the App-Lounge cache, and storage in e/OS settings.
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I restarted the phone
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I went into App-Lounge and then went straight to the ‘settings’ menu in App-Lounge. I unticked boxes ‘Automatically install updates’ and ‘Update Apps installed by other stores’ as the user aibd suggested above.
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I restarted the phones again.
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I went back into App-Lounge and went to updates. All the apps that I had installed with App-Lounge previously were there to be updated, and updated successfully.
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I went back into the App-Lounge ‘settings’ menu in the App-Lounge app. I reticked the option ‘Update Apps installed by other stores’
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I went back to ‘Updates’ in App-Lounge, and lo and behold all of the apps that I had installed manually from Github and the original developer’s websites were there, and I was able to update all of them successfully with App-Lounge.
This worked for both my phones the Moto Edge 20 (Berlin) and the Moto G 5G plus (nairo).
Conclusion: Problem solved! Thanks everyone for your help, you guys are awesome!