Hey Evanfromgbr,
Thanks for asking your question here in the Apple Communities. It looks like you are attempting o download your mail from Hotmail, while leaving the original messages on the Hotmail Server. This is the procedure for any mail account that uses the IMAP account settings. I will need you to visit this link, and enter your email address to verify your hotmail account is running IMAP, and not POP: Mail Settings Lookup
Once you verify you are running IMAP in your Hotmail account, we can then check a few settings to ensure your mail is being downloaded while you are connected to the internet, leaving a copy up on the server, and available for you to view if you are not currently connected to the internet. Please open Mail, and then click the word Mail up top next to the Apple. Click preferences.
We need to check the "General" tab, the tab on the far left. The second box down is called "Check for new messages:" You can set this as often as you like, leave it as Automatic, or change it to Manually. If you spend large amounts of time offline, to conserve battery power, you may wish to choose Manually. This is where you can click the option you seek, which will check for new mail and download them. To manually check for email, click the Get Mail
button. This envelope icon is at the top left of the main mail window, under the red, yellow and green dots for the window control, and is shown in our article: Use Mail on your Mac
I would like to remind you that when mail downloads from an IMAP account, a copy stays in the Hotmail servers, whether it downloads automatically, or if you choose the manual option. If you are running a POP account for your Hotmail, once the messages download to the device (be an iPhone, iPad, or Mac computer), they are no longer stored on the server. You would need to call Hotmail and ask to have your account changed from a POP account, to an IMAP if that is possible. Hope this help you with your mail.
Cheers.