I'm not sure this is a great sentence to translate as it is an idiom, and thus if you've never seen it before there is no hope of using the grammar and vocabulary to translate it.
I absolutely agree, but I guess I know what it means now that I got it wrong.
The idiom is "ein Auge zudrücken". 'doch' ist used to support the plea to do so, similar to something like "Please do bend the rules". Instead of 'doch' 'noch einmal' is more commonly used here.
In Italian we say "chiudere un occhio" to ignore some small irregularities. Is it the same in German?
Yes. You've been driving a little too fast and nothing bad happened? You might plead with the officer to ignore this minor misconduct. You're always a little sleepy in class because you study so hard until late at night? If you explained that to your teacher, he might not call your parents and give you a chance to improve your behaviour.
It would be more like "this one time": Can't you bend the rules, just this one time? – Können Sie nicht noch einmal ein Auge zu drücken? Or if you wanted to emphasise the character of exception: Können Sie nicht ein Auge zudrücken, nur dieses eine Mal? The latter being very close to my english suggestion.