There are already a number of blocks available to add this functionality to the user MyHome page, such as the Ajax Marking block and others.
https://moodle.org/plugins/view.php?plugin=block_ajax_marking
https://moodle.org/plugins/view.php?plugin=block_fn_marking
You add the block to your 'My' Page and it will show you all the assignments you need to mark. You can even make the My page the user default home page when they log in if required.
The workflow of the tutor may be different in different situations/institutions, and while it may not be perfect, Moodle does provide alternatives to allow you to customise the features and add plugins to make it better for your particular situation if you can get your admins to add the relevant plugins to your system.
Richard