@Vero
Hello, I am having the same problem with Alfred disappearing and behaving the same way.
However, it only happens intermittently when attaching a secondary 4K monitor via USB-C. If I unplug the monitor, Alfred will start appearing again. Plug it back in, it will intermittently decide to show up again.
The only way I have found to resolve the issue reliably when it happens is to quit and re-open Alfred, and it begins working again.
I've been using Alfred for years and this issue is new to Catalina.
I'm running macOS 10.15.2 and Alfred 4.0.7 [1131].