When doing it with a smart plug, your fan needs to be
really dumb, with only a knob or mechanical buttons to set power
If it beeps or lights up or has electronic buttons, chances are it won't work Since the default "power up" state is probably off
Before you go too far, you want to test to make sure your fan will work.
To test this, have the fan running, and unplug it from the wall, wait a bit (20 seconds to a couple minutes depending on how bored you are) and plug it back in. that's basically what the smart switch will do. if the fan doesn't start spinning when you plug it in, it's not going to work.
to my knowledge there is no way to cycle based on time, (on for 5 minutes, off for 10) you might be able to do it in a shortcut, but I believe currently there are issues with longer times (anything over a couple minutes) I think I saw this was addressed in iOS14.
you can do time based automations in homekit, on at 1am, off at 1:30am, but you would have to set up each time individually.
if you have a homekit temperature sensor, you can automate the switch based on the temperature in the room. But you can't do that in the home app. You need to get the Eve app, it's free, and has a lot more automation features than home. The eve app works with all homekit devices, not just those from eve.
I agree we need a pic of your switch. It should only have a plug to go into the outlet in the wall, an outlet to plug whatever you want to control. and possibly a button to control it locally
these 2 should work, I think you're in the UK, you might be able to find cheaper elsewhere. And there are other options, these are just available from apple, and have a good reputation. They also don't require a hub, THey'll connect via bluetooth to your HomePod, or phone.
smart plug -
https://www.apple.com/uk/shop/product/HKEH2B/A/eve-energy
temp sensor -
https://www.apple.com/uk/shop/product/HMLY2ZM/A/eve-room-indoor-air-quality-monitor
