To the OP, the info. and spec's (and name!) of the device given in #13 above all suggest this device is one whose main function is working with cloud-based services and storage (hence the low onboard storage size and generally low hardware specifications). The Windows 10 S mode configuration looks like it is the MS offering to counter the Chromebook (a 'lite' operating system on lite computer hardware).
If you bought the device hoping it would work/perform as a normal laptop (esp. for things like storing your own data on the device itself) then this was probably the wrong device for your needs.
I can't find any devices on Argos now which match the spec's given. However, on the Argos website they do list some similar devices just as 'Microsoft Windows 10' whereas a closer look into the spec's on the manufacturer's website shows more detail as 'Windows 10 Home in S Mode'.
Based on that info. you could possibly argue that was not clear as to what exactly it was that you were getting (with S Mode being a more restricted version of the OS). You may, therefore, have some grounds for redress via part refund or exchange if your device was actually advertised in that same way.
Earlier in the year I was asked to help with different HP laptop (quite high spec) which had graphics problems on AMD/Radeon where the display also frequently kept crashing. Unfortunately, never resolved it. Seemed to be a very common issue with that particular manufacturer/processor/graphics configuration on that laptop type.
Sounds like you have had a few prob's with the device so you should refer back to Argos under the Consumer Rights Act 2015 as I believe the retailer is your first point of contact (even if they try to fob you off with a manufacturer's warranty route). Things are a bit more problematic as you are nearing one year of ownership but you should probably give it a go anyway.
Thank you Goslow. There was nothing in the description which suggested any of what you said. I know nothing about computers so thought 30 GB was a lot of storage. Since the Windows update a while back, I haven't stored anything on the laptop but it still shuts down.
I managed to engage with someone on the Argos chat again this afternoon. She didn't seem interested in what was wrong, just told me to take it back to an Argos store (but not a Sainsburys store which is a 15 minute walk away) and ask them about it. My nearest store is 2 bus rides away as I don't have a car but they said I could take it later. Not sure when that'll be as I won't feel safe using 4 busses in one day. I need to kick up a bit more. lol