"Please do not clean shoes with a handkerchief."
The verb is 磨く(みがく). To make the negative conjugation for verbs that end in くchange it to かない。 That makes it 磨かない. By the way, 磨けない is the negative potential so your sentence would mean "Please don't be able to polish your shoes with a handkerchief ".
I think it is good to change to 'みがかない' from 'みがけない' in this sentence. The third letter of 'みがかない'.