The key phrase there Irith is 'After they submit their assignment'
To allow them to edit it the teacher would have to revert it back to draft
But the whole point is that once its submitted, its like handing in a piece of paper to the teacher, they can't edit it unless the teacher gives it back to them to do some more work on.
If you want them to be able to upload a piece of work and continue to work on it, then make sure the option to have the students click the submit for marking button is enabled, that way they can use the assignment submission as a 'file store' until they are happy they have everything ready and then click submit to actually hand it in.