When GMX has been looking for mail on your Tesco.net account it's then deleted them, meaning that they're then held solely on GMX's server and not on Tesco's.
Log into your GMX account via their web page:
http://www.gmx.co.uk/
Click on 'E-mail Settings' (bottom left)
Click on 'Mail Collector'
Remove the tick alongside 'Delete e-mails from the third party server after fetching'
Click 'Save'
Problem solved.